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Solution components in a degenerate weighted BVP
Nonlinear Analysis 192 (2020) 111690
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Solution components in a degenerate weighted BVP✩ D. Aleja a , I. Antón b , J. López-Gómez c ,∗ a
Departamento de Matemática Aplicada, Ciencia e Ingeniería de los Materiales y Tecnología Electrónica, Universidad Rey Juan Carlos, Móstoles (Madrid) 28933, Spain b Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, Madrid 28040, Spain c Instituto de Matemática Interdisciplinar (IMI), Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, Madrid 28040, Spain
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Article history: Received 22 July 2019 Accepted 4 November 2019 Communicated by Vicentiu D. Radulescu MSC: primary 35K57 35B09 35J25
abstract This paper ascertains the structure of the set of positive solutions of a degenerate weighted logistic equation with a continuous kinetic under non-classical mixed boundary conditions. After establishing the uniqueness of the positive solution and the existence of two global components of positive solutions, it is shown that any component must be a continuous curve. Further, some general sufficient conditions are provided, in terms of the weight functions involved in the setting of the model, so that it possesses exactly two components of positive solutions. The problem of finding out the total number of components in the general case remains open. © 2019 Elsevier Ltd. All rights reserved.
Keywords: Generalized logistic equation Weighted problem Degenerate equation Non-classical mixed boundary conditions Structure of the positive solutions Counting the number of components
1. Introduction This paper studies the positive steady-states of the semilinear parabolic problem ⎧ in Ω × (0, ∞), ⎨ ∂t u + Lu = λm(x)u − a(x)h(u)u Bu = 0 on ∂Ω × (0, ∞), ⎩ u(·, 0) = u0 in Ω ,
(1.1)
where Ω is a bounded subdomain (open and connected set) of class C 2 of RN , N ≥ 1, L is a second order uniformly elliptic operator L := −div (A∇·) + ⟨b, ∇·⟩ + c, (1.2) ✩ This paper has been supported by the IMI of Complutense University and the Ministry of Education, Science and Universities of Spain under Research Grant PGC2018-097104-B-100. ∗ Corresponding author.
E-mail addresses: [email protected] (D. Aleja), [email protected] (I. Antón), [email protected] (J. López-Gómez). https://doi.org/10.1016/j.na.2019.111690 0362-546X/© 2019 Elsevier Ltd. All rights reserved.
D. Aleja, I. Antón and J. López-Gómez / Nonlinear Analysis 192 (2020) 111690
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with 1,∞ A ∈ Msym (Ω )), N (W
N
b ∈ (L∞ (Ω )) ,
and B is a non-classical mixed boundary operator { ξ on Γ0 , Bξ := ∂ν ξ + β(x)ξ on Γ1 ,
c ∈ L∞ (Ω ),
ξ ∈ C(Γ0 ) ⊕ C 1 (Ω ∪ Γ1 ),
(1.3)
where Γ0 and Γ1 are two disjoint open and closed subsets of ∂Ω such that ∂Ω := Γ0 ∪ Γ1 , β ∈ C 1 (Γ1 ), and ν := An is the co-normal vector field; n standing for the outward normal vector-field to Ω . Throughout this paper, for any pair of real functions h1 , h2 in Ω , it is said that h1 ⪈ h2 if h1 ≥ h2 but h1 ̸= h2 . The standard notation h1 > h2 might be slightly confusing, for as it could suggest that h1 (x) > h2 (x) for all x ∈ Ω . In ¯ ) is a continuous function that changes sign in the model (1.1), λ ∈ R is regarded as a parameter, m ∈ C(Ω ¯ ¯ Ω , a ∈ C(Ω ) satisfies a ⪈ 0 and the nonlinearity h ∈ C(R) is assumed to satisfy h(0) = 0,
h(u) > h(v) > 0 if u > v > 0, and
lim h(u) = ∞.
u→∞
As in applications to population dynamics the problem (1.1) models the evolution of a dispersing species, u, in the inhabiting territory Ω (see, e.g., Aleja & L´ opez-G´omez [1–3], as well as the monographs of Murray [18] ∞ and L´ opez-G´ omez [17]), the initial data, u0 ∈ L (Ω ), is assumed to satisfy u0 ⪈ 0. Under these assumptions, (1.1) possesses a unique classical solution, u(t, x; u0 ), which is strongly positive for all t > 0, and the dynamics of (1.1) is governed either by the non-negative solutions of the semilinear elliptic boundary value problem { Lw = λm(x)w − a(x)h(w)w in Ω , (1.4) Bw = 0 on ∂Ω , or by the underlying metasolutions of (1.4) (see, e.g., Fraile et al. [12] and Daners & L´opez-G´omez [8]). The main goal of this paper is ascertaining the global structure of the set of positive solutions of (1.4), (λ, w), w ⪈ 0, according to the nature of the weight functions m, a, and the size of the parameter λ ∈ R. This task is accomplished in two particular cases when the principal eigenvalue of the term (L − λm, B, Ω ), denoted throughout this paper by Σ (λ) ≡ σ[L − λm, B, Ω ],
λ ∈ R,
(1.5)
satisfies Σ (λ0 ) > 0 for some λ0 ∈ R. As it will be shown in Section 2, under this assumption, Σ −1 (0) = {λ− , λ+ } with λ− < λ0 < λ+ , and the set of solutions of (1.4) possesses two unbounded components, C− and C+ , bifurcating from zero at (λ− , 0) and (λ+ , 0), respectively, consisting of positive solutions of (1.4), except for the bifurcation points (λ± , 0). Besides generalizing very substantially all previous existence and uniqueness results up to cover the most general case when h(w) is merely continuous and β(x) changes of sign, so substantially generalizing the pioneering ones of J. M. Fraile et al. [12], this paper addresses the extremely challenging problem of ascertaining the exact number of components of the set of positive solutions of (1.4). Actually, our main aim is establishing that C+ and C− are the unique components of positive solutions of (1.4) under each of the following structural conditions on a−1 (0): (A) Ω0 ≡ int a−1 (0) ̸= ∅ is an open subset of class C 2 of Ω with finitely many components, Ω0j , 1 ≤ j ≤ q, ¯i ∩ Ω ¯ j = ∅ if i ̸= j. such that Ω 0 0 −1 (B) a (0) ⊂ K− ∪ K+ , where K− and K+ are two disjoint compact subsets of Ω such that m(x) < 0 for all x ∈ K− and m(x) > 0 for all x ∈ K+ . The condition (A) extends the condition of J. M. Fraile et al. [12] to cover the case when Ω0 consists of finitely many smooth components, while (B) does not impose any restriction on a−1 (0), except that m cannot vanish on it. From a technical point of view, in order to solve this problem, this paper contributes with the next three technical novelties, which might be useful in a variety of contexts:
D. Aleja, I. Antón and J. López-Gómez / Nonlinear Analysis 192 (2020) 111690
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(i) In the regions where a(x) > 0, among them the edges of Ω , ∂Ω , the positive constants cannot provide us with supersolutions of (1.4) if β is somewhere negative. We can overcome this technical difficulty by using an exponential function, eM ψ , with sufficiently large M > 0, in order to absorb the negative ¯ ) is a smooth function with min∂Ω ∂ν ψ > 0. Such a part of β with the constant M . Here, ψ ∈ C 2 (Ω function exists by [16, Lem. 2.1]; its regularity equals the regularity of Ω (see Li & Nirenberg [13] and Fern´ andez-Rinc´ on & L´ opez-G´ omez [11] for any further required details). ˜ w), ˜ > λ+ (resp. λ ˜ < λ− ), must lie on (ii) The proof of the fact that any positive solution of (1.4), (λ, ˜ with λ C+ (resp. C− ) under condition (A), or (B), relies on the uniqueness of the positive solution established by Theorem 3.1 through the construction of a certain supersolution to prove that (1.4) has a positive ˜ (resp. λ ∈ [λ, ˜ λ− )). Each of the conditions (A) and (B) guarantees the solution for every λ ∈ (λ+ , λ] existence of such a supersolution. Although its construction is folklore when m has constant sign, it is far from obvious when m changes of sign. These constructions are extremely sharp because (1.4) cannot admit any positive solution if λ ∈ [λ− , λ+ ]. Both use the supersolutions eM ψ within the patches of Ω where a(x) > 0. (iii) This paper provides a (non-trivial) general method to prove that a component of the solution set must be a continuous curve if there is uniqueness. The method involves a tricky use of the uniqueness, the compactness of the associated resolvent operator, and the fact that the components are maximal closed and connected subsets. In the existing literature a global continuation method based on the implicit function theorem is commonly used to ascertain the structure of the solution set. As in this paper h(w) is far from smooth, the implicit function theorem cannot be invoked. This paper is organized as follows. Based on the global bifurcation results of [15], Section 2 establishes the existence of C− and C+ . Section 3 shows the uniqueness of the positive solution and infers from it that C+ and C− are continuous curves. Sections 4 and 5 show that these are the unique components of positive solutions under conditions (A) and (B), respectively. Finally, in Appendix we provide an abstract existence result that we could not find elsewhere. It is a refinement of a classical result of Amann [4]. Although the existence of monotone sequences starting from a couple of ordered sub and super solutions, u ≤ v, and converging respectively to the minimal and the maximal solutions of the problem in the interval [u, v] was proven without the restrictions of our Theorem 6.1 by C. De Coster and P. Omari in [9, Le. 2.7], we have preferred to deliver a self-contained proof of the precise result needed in this paper, Theorem 6.1, because [9] focused attention on the case of homogeneous Dirichlet boundary conditions, while here we are dealing with a non-classical mixed boundary operator. The existence of the minimal and the maximal solutions might be also established by adapting the topological argument of J. A. Cid [6]. 2. The two components bifurcating from (λ, 0) Throughout this paper, for any given term (L, B, Ω ), as those introduced in Section 1, σ[L, B, Ω ] stands for the principal eigenvalue of (L, B, Ω ), i.e., the unique value of τ for which the linear eigenvalue problem { Lφ = τ φ in Ω , Bφ = 0 on ∂Ω , admits a positive eigenfunction φ ⪈ 0. In such case, by e.g. [16, Ch. 7], φ ≫ 0, in the sense that φ(x) > 0 for all x ∈ Ω ∪ Γ1 and ∂ν φ(x) < 0 if x ∈ Γ0 . Also, we set Σ (λ) ≡ σ[L − λm, B, Ω ],
λ ∈ R.
(2.1)
4
D. Aleja, I. Antón and J. López-Gómez / Nonlinear Analysis 192 (2020) 111690
Fig. 1. The graph of Σ (λ).
This principal eigenvalue plays a pivotal role to characterize the existence of positive solutions of (1.4). According to either [16, Th. 9.1], or [14], Σ (λ) is analytic, strictly concave and, since m(x) changes of sign in Ω , lim Σ (λ) = −∞. λ→±∞
Therefore, if Σ (λ0 ) > 0 for some λ0 ∈ R, then ⎧ ⎨ <0 =0 Σ (λ) ⎩ >0
there exist λ− < λ0 < λ+ such that iff λ ∈ (−∞, λ− ) ∪ (λ+ , ∞), iff λ ∈ {λ− , λ+ }, iff λ ∈ (λ− , λ+ ).
(2.2)
Fig. 1 shows the graph of Σ (λ) in this case. The main goal of this section is establishing the existence of two components of the set of positive solutions of (1.4) emanating from the curve of trivial solutions (λ, 0) at λ = λ+ and λ = λ− . The next result establishes a necessary condition for the existence of a positive solution of (1.4). It reveals the importance of the sign of Σ (λ) in our analysis. Lemma 2.1.
If (1.4) admits a positive solution, w, then Σ (λ) < 0. Moreover, σ[L − λm + ah(w), B, Ω ] = 0
(2.3)
and w ≫ 0 in the sense that w(x) > 0 for all x ∈ Ω ∪ Γ1 and ∂ν w(x) < 0 if x ∈ Γ0 . Proof . Let w be a positive solution of (1.4). Then, w ⪈ 0 and { (L − λm + ah(w))w = 0 in Ω , Bw = 0 on ∂Ω , i.e., w is a positive eigenfunction associated with the eigenvalue zero of the term (L − λm + ah(w), B, Ω ). Thus, by the uniqueness of the principal eigenvalue, (2.3) holds and w ≫ 0. Consequently, owing to the monotonicity of the principal eigenvalue with respect to the potential established by S. Cano-Casanova and J. L´ opez-G´ omez in [5, Prop. 3.3], it is apparent that 0 = σ[L − λm + ah(w), B, Ω ] > σ[L − λm, B, Ω ] = Σ (λ), because ah(w) ⪈ 0. This ends the proof. □
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The next result establishes that the positive solutions of (1.4) can only bifurcate from (λ, 0) at the values of the parameter where Σ (λ) = 0. Lemma 2.2. Let λs ∈ R be such that (λ, w) = (λs , 0) is a bifurcation point to positive solutions of (1.4) from (λ, 0). Then, Σ (λs ) = 0. In particular, λs ∈ {λ− , λ+ } if Σ (λ0 ) > 0 for some λ0 ∈ R. Proof . By elliptic regularity, for every κ > −σ[L, B, Ω ], the positive solutions of (1.4) are the zeros (λ, w), ¯ ) → C(Ω ¯ ) defined by with w ⪈ 0, of the operator F : R × C(Ω F(λ, w) := w − (L + κ)−1 [(λm + κ − ah(w))w].
(2.4)
¯ ) such that wn ⪈ 0, Let (λn , wn ), n ≥ 1, be a sequence in R × C(Ω lim (λn , wn ) = (λs , 0) and F(λn , wn ) = 0 for all n ≥ 1.
n→∞
Then, dividing by ∥wn ∥C(Ω) ¯ , n ≥ 1, we have that [ ] wn wn −1 = (L + κ) (λn m + κ − ah(wn )) , ∥wn ∥C(Ω) ∥wn ∥C(Ω) ¯ ¯ Note that the sequence Hn := (λn m + κ − ah(wn ))
wn , ∥wn ∥C(Ω) ¯
n ≥ 1.
(2.5)
n ≥ 1,
¯ ), because is bounded in C(Ω ∥Hn ∥C(Ω) ¯ ≤ |λn |∥m∥C(Ω) ¯ + κ + ∥ah(wn )∥C(Ω) ¯ and lim
n→∞
(
) |λn |∥m∥C(Ω) = |λs |∥m∥C(Ω) ¯ + κ + ∥ah(wn )∥C(Ω) ¯ ¯ + κ.
¯ ) → C(Ω ¯ ), there exists a subsequence of Thus, by the compactness of the resolvent operator (L + κ)−1 : C(Ω wn /∥wn ∥C(Ω) ¯ , relabeled by n, such that lim
n→∞
wn =φ ∥wn ∥C(Ω) ¯
¯ ) for some φ ∈ C(Ω ¯ ). Necessarily, ∥φ∥ ¯ = 1 and φ ⪈ 0. Moreover, letting n → ∞ in (2.5) yields in C(Ω C(Ω) φ = (L + κ)−1 (λs m + κ)φ. By elliptic regularity, φ must be a strong solution of { (L − λs m)φ = 0 Bφ = 0
in Ω , on ∂Ω .
Consequently, since φ ⪈ 0, Σ (λs ) = 0 holds. This ends the proof.
□
The next result is the main theorem of this section. By a component it is meant a closed and connected subset that is maximal for the inclusion. Theorem 2.1. Suppose that Σ (λ0 ) > 0 for some λ0 ∈ R. Then, the set of non-negative solutions of (1.4) possesses two components, C+ and C− , such that (λ± , 0) ∈ C± , w ≫ 0 if (λ, w) ∈ C± \ {(λ± , 0)}, and C± ¯ ). Moreover, according to Lemma 2.1, are unbounded in R × C(Ω C− ⊂ {(λ, w) : λ < λ− , w ≫ 0} ∪ {(λ− , 0)}, C+ ⊂ {(λ, w) : λ > λ+ , w ≫ 0} ∪ {(λ+ , 0)}.
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D. Aleja, I. Antón and J. López-Gómez / Nonlinear Analysis 192 (2020) 111690
Proof . Throughout this proof, for any fixed κ > −σ[L, B, Ω ], we consider the operators L(λ)w := w − (L + κ)−1 [(λm + κ)w],
N(w) := (L + κ)−1 (ah(w)w).
Since L(λ) is a compact perturbation of the identity map, it is a real analytic family of Fredholm operators of index zero. Moreover, N(w) ¯) lim = 0 in C(Ω w→0 ∥w∥C(Ω) ¯ and, due to (2.4), F(λ, w) = L(λ)w + N(w) satisfies the general structural assumptions of [15, Ch. 7]. By Lemma 2.2, λ± are the unique possible bifurcation values to positive solutions of (1.4) from (λ, 0). Moreover, owing to [16, Ch. 7], if φ± ≫ 0 stands for a principal eigenfunction associated to Σ (λ± ) = 0, then N [L(λ± )] = span {φ± } . On the other hand, since Σ (λ0 ) = σ[L − λ0 m, B, Ω ] > 0 and the principal eigenvalue is dominant, 0 cannot be an eigenvalue of (L − λ0 m, B, Ω ), which implies ¯ )). Therefore, according to [15, Sect. 4.4], the set of eigenvalues of L(λ), L(λ0 ) ∈ Iso(C(Ω Λ = {λ ∈ R : dim N [L(λ)] ≥ 1} , is discrete. Consequently, the algebraic multiplicity of Esquinas and L´opez-G´omez [10] of L(λ) at λ = λ± , χ[L(λ); λ± ], is well defined. Actually, χ[L(λ); λ+ ] = χ[L(λ); λ− ] = 1,
(2.6)
because λ− and λ+ are simple eigenvalues of (L(λ), L′ (λ)) as discussed by Crandall and Ravinowitz [7], i.e., L′ (λ± )φ± ∈ / R[L(λ± )]. Indeed, on the contrary, assume that L′ (λ± )φ± = −(L + κ)−1 (mφ± ) ∈ R [L(λ± )] . ¯ ) such that Then, there exists v± ∈ C(Ω v± = (L + κ)−1 (λ± mv± + κv± − mφ± ) or, equivalently, {
(L − λ± m)v± = −mφ± Bv = 0
in Ω , on ∂Ω .
But, thanks to [16, Th. 9.4], this is not possible because mφ± ∈ / R[L − λ± m]. Thus, (2.6) holds and, hence, due to [15, Th. 5.6.2], the Leray–Schauder degree of L(λ) in BR (0), R > 0, changes by 1 as λ crosses λ+ and λ− . Therefore, by [15, Th. 6.4.3 and Lem. 6.5.4], there exist two components of the set solutions of (2.5), C+ and C− , such that (λ± , 0) ∈ C± and w ≫ 0 if (λ, w) ∈ C± \ {(λ± , 0)}. Moreover, each of these components satisfies some of the next alternatives: ¯ ). (A1) C± is unbounded in R × C(Ω (A2) There exists µ± ∈ Λ \ {λ± } such that (µ± , 0) ∈ C± .
D. Aleja, I. Antón and J. López-Gómez / Nonlinear Analysis 192 (2020) 111690
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(A3) C± contains a point (ρ± , y± ) ∈ R × (Y± \ {0}), where ∫ { } ¯ Y± := y ∈ C(Ω ) : yφ± = 0 . Ω
Note that ¯ ) = Y± ⊕ span {φ± } . C(Ω ¯ ). So, Y± is a supplement of N [L(λ± )] in C(Ω According to Lemma 2.2, the alternative (A2) cannot occur, since (λ± , 0) ∈ C± and, due to Lemma 2.1, (1.4) cannot admit a positive solution if λ ∈ (λ− , λ+ ). So, (λ∓ , 0) ∈ / C± . Alternative (A3) cannot occur neither, because the nontrivial solutions on Y± cannot be positive. Therefore, the alternative (A1) holds. The proof is complete. □ 3. The components C+ and C− are continuous curves The next result establishes the uniqueness of the positive solution of (1.4) if it exists. Theorem 3.1.
Suppose that (1.4) admits a positive solution for some λ ∈ R. Then, it is unique.
Proof . By Lemma 2.1, Σ (λ) < 0. Suppose that (1.4) admits two positive solutions, w ̸= v. Then, { (L − λm + ah(w))w = (L − λm + ah(v))v = 0 in Ω , Bw = Bv = 0 on ∂Ω .
(3.1)
Moreover, w ≫ 0, v ≫ 0 and σ[L − λm + ah(w), B, Ω ] = σ[L − λm + ah(v), B, Ω ] = 0. On the other hand, since h(Cw) ≥ h(w) for all C ≥ 1, it follows that (L − λm + ah(Cw))Cw ≥ C(L − λm + ah(w))w = 0, and hence, w := Cw provides us with a supersolution of (1.4) for all C ≥ 1. Now, let φ ≫ 0 be the unique principal eigenfunction associated to Σ (λ) normalized so that ∥φ∥C(Ω) ¯ = 1. Then, w := εφ provides us with a positive subsolution of (1.4) for sufficiently small ε > 0. Indeed, as for sufficiently small ε > 0 Σ (λ) ≤ Σ (λ) + ah(ε) < 0, it is apparent that (L − λm + ah(w))w = (Σ (λ) + ah(εφ))w ≤ (Σ (λ) + ah(ε))w ≤ 0. Thus, shortening ε > 0 and enlarging C > 1, if necessary, we can assume that 0 ≪ w = εφ ≤ min{w, v} ⪇ max{w, v} ≤ Cw = w, and hence, by Theorem 6.1 in the Appendix, which is a refinement of the main theorem of Amann [4], it becomes apparent that the minimal positive solution of (1.4) in the order interval [w, w], say w∗ , satisfies 0 ≪ w ≤ w∗ ≤ w
and
0 ≪ w ≤ w∗ ≤ v.
Therefore, without lost of generality, we can assume that v ⪇ w. Suppose that ah(v) = ah(w)
in Ω .
(3.2)
D. Aleja, I. Antón and J. López-Gómez / Nonlinear Analysis 192 (2020) 111690
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Then, 0 = (L − λm)(w − v) + a[h(w)w − h(v)v] = (L − λm + ah(v))(w − v). Thus, since w − v ⪈ 0 and σ[L − λm + ah(v), B, Ω ] = 0, by the strong positivity of the principal eigenfunction, w ≫ v. Hence, in the support of a(x), it is apparent that ah(v) < ah(w), which contradicts (3.2). Therefore, since v ⪇ w, ah(w) ⪈ ah(v)
in Ω .
So, 0 = σ[L − λm + ah(w), B, Ω ] > σ[L − λm + ah(v), B, Ω ] = 0, which is a contradiction. Consequently, (1.4) cannot admit two positive solutions. □ Through the rest of this paper, Pλ stands for the λ-projection operator, Pλ (λ, w) = λ. As Pλ is a continuous map and the components C± are connected, Pλ (C− ) and Pλ (C+ ) are connected in R. Thus, since (λ± , 0) ∈ C± and, owing to Lemma 2.1, Pλ (C± ) ∩ (λ− , λ+ ) = ∅, it becomes apparent that there exist λ∗− ∈ [−∞, λ− ) and λ∗+ ∈ (λ+ , ∞] such that { } Pλ (C− ) ∈ (λ∗− , λ− ], [λ∗− , λ− ]
{ } and Pλ (C+ ) ∈ [λ+ , λ∗+ ), [λ+ , λ∗+ ] .
The next result establishes that each of the components C+ and C− has the global structure of a continuous curve bifurcating from (λ, 0) at λ = λ+ and λ = λ− , respectively. So, both components are connected by arcs. Subsequently, for every λ ∈ Pλ (C± ) \ {λ± }, we will denote by w(λ) the unique positive solution of (1.4), and we set w(λ± ) := 0. Theorem 3.2.
¯ ), λ ↦→ w(λ), is continuous. Therefore, The map Pλ (C± ) → C(Ω C± = {(λ, w(λ)) : λ ∈ Pλ (C± )}
are continuous curves bifurcating from (λ, 0) at λ = λ± . Consequently, Pλ (C− ) = (λ∗− , λ− ],
Pλ (C+ ) = [λ+ , λ∗+ ),
and lim ∥w(λ)∥C(Ω) ¯ =∞
λ→λ∗ ±
if |λ∗± | < +∞.
(3.3)
D. Aleja, I. Antón and J. López-Gómez / Nonlinear Analysis 192 (2020) 111690
9
Proof . To prove the continuity of w(λ) in λ it suffices to show that lim w(λn ) = w(λω )
n→∞
for any infinity sequence {λn }n≥1 of Pλ (C± ) such that lim λn = λω ∈ Pλ (C± ).
(3.4)
n→∞
As the proof is the same for both components, C± , we will detail it only for C+ . Our proof proceeds by contradiction. So, assume that there is a sequence {λn }n≥1 of Pλ (C+ ) such that (3.4) holds for some λω ∈ Pλ (C+ ) and lim ∥w(λn ) − w(λω )∥C(Ω) ¯ ̸= 0. n→∞
Then, there exist R > 0 and a subsequence of {λn }n≥1 , relabeled by n ≥ 1, such that ∥w(λn ) − w(λω )∥C(Ω) ¯ >R
for all n ≥ 1.
(3.5)
Suppose that w(λω ) is a positive solution of (1.4) and note that, according to Lemma 2.1, this entails that λω > λ+ . As λn , λω ∈ Pλ (C+ ) for all n ≥ 1, by the uniqueness of the positive solution established by Theorem 3.1, it follows from the definition of w(λ) that (λω , w(λω )), (λn , w(λn )) ∈ C+ for all n ≥ 1. ¯ ), it follows from (3.5) that, for every n ≥ 1, there exists µn ∈ Pλ (C+ ) Thus, since C+ is connected in R×C(Ω such that ∥w(µn ) − w(λω )∥C(Ω) (3.6) ¯ = R. We claim that µn can be chosen so that |µn − λω | ≤ |λn − λω |.
(3.7)
To prove this, we will argue by contradiction assuming that ∥w(µ) − w(λω )∥C(Ω) ¯ ̸= R
if µ ∈ Pλ (C+ ) satisfies |µ − λω | ≤ |λn − λω |.
(3.8)
Note that λn = ̸ λω for all n ≥ 1, as in the contrary case, we would contradict (3.5). Suppose that λn > λω , as illustrated in Fig. 2, and consider ˆ n, µ ˆn := inf M
ˆ n := {ρ ∈ Pλ (C+ ) ∩ (λn , +∞) : ∥w(ρ) − w(λω )∥ = R}. M
ˆ n ̸= ∅. Indeed, if M ˆ n = ∅, then, by (3.8), µn < λω and, since λω < λn , (1.4) would possess Note that M two positive solutions for λ = λω : one in the interior of the ball centered at w(λω ), itself, and another one ˆ n ̸= ∅ and µ in the exterior, which again contradicts Theorem 3.1. Thus, M ˆn is well defined. Moreover, by definition, µ ˆn ≥ λn . Next, we will show that actually µ ˆn > λn . The proof will proceed, again, by contradiction. Assume that ˆ n , ℓ ≥ 1, a sequence such that µ ˆn = λn for some n ≥ 1, and let ρℓ ∈ M lim ρℓ = µ ˆ n = λn ,
ℓ→∞
∥w(ρℓ ) − w(λω )∥ = R,
ℓ ≥ 1.
(3.9)
Thanks to (3.9), (ρℓ , w(ρℓ )), ℓ ≥ 1, provides us with a bounded sequence of positive solutions of (1.4) in ¯ ). Now, remember that the positive solutions of (1.4) are the zeros of the operator F defined in (2.4). C(Ω Thus, for every ℓ ≥ 1, w(ρℓ ) = (L + κ)−1 Hℓ where Hℓ := [ρℓ m + κ − ah(w(ρℓ ))]w(ρℓ ),
10
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Fig. 2. The construction of µ ˆn .
¯ ), it follows from the compactness of the operator (L + κ)−1 : C(Ω ¯ ) → C(Ω ¯) and, since Hℓ is bounded in C(Ω that there exists a subsequence of w(ρℓ ), labeled again by ℓ, such that lim w(ρℓ ) = w
ℓ→∞
¯ ). Moreover, letting ℓ → ∞ yields in C(Ω w(ρℓ ) = (L + κ)−1 Hℓ −→ w = (L + κ)−1 [(λn m + κ − ah(w))w]. Thus, by elliptic regularity, (λn , w) is a strong non-negative solution of (1.4). Moreover, w ̸= w(λn ) because ∥w − w(λω )∥C(Ω) ¯ = R,
∥w(λn ) − w(λω )∥C(Ω) ¯ > R,
which contradicts the uniqueness of the positive solution established by Theorem 3.1. Consequently, µ ˆn > λn . Therefore, for every λ ∈ (λn , µ ˆn ), (1.4) has, at least, two positive solutions: one in BR and another one in ¯) \ B ¯R , which again is impossible by Theorem 3.1. It should be noted that C+ cannot link (λω , w(λω )) C(Ω to (λn , w(λn )) through some (ρ, w(ρ)) with ρ < λω ,
∥w(ρ) − w(λω )∥C(Ω) ¯ = R,
because w(λω ) is the unique positive solution of (1.4) at λω . This argument can be easily adapted to get another contradiction in the case when λn < λω by considering ˜ n, µ ˜n := sup M
˜ n := {ρ ∈ Pλ (C+ ) ∩ (−∞, λn ) : ∥w(ρ) − w(λω )∥ = R}, M
but we refrain ourselves of giving any further details by repetitive. This contradiction shows that the µn ’s can be chosen to satisfy (3.7). According to (3.4) and (3.7), we can infer that lim µn = λω .
n→∞
(3.10)
¯ ). Moreover, thanks to (3.6), (µn , w(µn )), n ≥ 1, is a bounded sequence of positive solutions of (1.4) in C(Ω Consequently, adapting the previous compactness argument, along some subsequence, relabeled by n, it can be easily seen that lim w(µn ) = w n→∞
D. Aleja, I. Antón and J. López-Gómez / Nonlinear Analysis 192 (2020) 111690
11
¯ ), where (λω , w) is a positive solution of (1.4) with in C(Ω ∥w − w(λω )∥C(Ω) ¯ = R. Therefore, (1.4) possesses two positive solutions at λ = λω , which, once more, contradicts Theorem 3.1. This contradiction shows that λω = λ+ and w(λ+ ) = 0. But in such case, according to Lemma 2.1, (1.4) cannot admit a positive solution, which contradicts the fact that (λ+ , w) is a positive solution. Note that, in this case, by construction, ∥w∥C(Ω) ¯ =R and hence w must be positive. This contradiction ends the proof of the continuity of the map λ ↦→ w(λ) and shows that indeed C+ and C− are continuous curves bifurcating from (λ, 0) at λ = λ+ and λ = λ− , respectively. Now, suppose, e.g., that Pλ (C+ ) = [λ+ , λ∗+ ] for some λ+ < λ∗+ < +∞. Then, since w(λ) is continuous in λ, C+ = {(λ, w(λ)) : λ ∈ [λ+ , λ∗+ ]} is a compact arc of continuous curve. In particular, C+ must be bounded, which contradicts Theorem 2.1. Therefore, Pλ (C+ ) = [λ+ , λ∗+ ) for some λ+ < λ∗+ ≤ +∞. (3.11) Similarly, it is easily seen that Pλ (C− ) = (λ∗− , λ− ] for some − ∞ ≤ λ∗− < λ− .
(3.12)
Finally, suppose that λ∗+ < +∞ in (3.11). Then, since for sufficiently small ε > 0 the set {(λ, w(λ)) : λ ∈ [λ+ , λ∗+ − ε]} is bounded, because it is a compact arc of continuous curve, it becomes apparent that lim sup ∥w(λ)∥C(Ω) ¯ = +∞ λ↑λ∗ +
¯ ). Similarly, since C+ is unbounded in R × C(Ω lim sup ∥w(λ)∥C(Ω) ¯ = +∞ λ↓λ∗ −
if λ∗− > −∞. The proof of (3.3) can be easily accomplished with an additional compactness argument within the vain of the previous one. Indeed, if (3.3) does not hold at, e.g., λ∗+ , then there exists a sequence λn , n ≥ 1, ¯ ). Thus, such that limn→∞ λn = λ∗+ with (λn , w(λn )) ∈ C+ and such that w(λn ), n ≥ 1, is bounded in C(Ω thanks to the previous compactness argument, along some subsequence, relabeled by n, we have that lim w(λn ) = w∗ ≥ 0
n→∞
where (λ∗+ , w∗ ) is a non-negative solution of (1.4). Since (λ∗+ , w∗ ) ∈ C+ and λ∗+ > λ+ , we find that w∗ ≫ 0 and therefore, λ∗+ ∈ Pλ (C+ ), which contradicts the fact that Pλ (C+ ) = [λ+ , λ∗+ ). This concludes the proof. □ Fig. 3 shows two admissible components C− and C+ in the special case when λ∗− and λ∗+ are real numbers. Since m(x) changes sign, the mapping λ ↦→ w(λ) might not be monotone.
D. Aleja, I. Antón and J. López-Gómez / Nonlinear Analysis 192 (2020) 111690
12
Fig. 3. An admissible global bifurcation diagram.
Fig. 4. An admissible case for hypothesis (A).
4. Under assumption (A) the solution set consists of C± \ {(λ± , 0)} Throughout this section, we assume that (A) Ω0 ≡ int a−1 (0) ̸= ∅ is an open subset of class C 2 of Ω with finitely many components, Ω0j , 1 ≤ j ≤ q, ¯i ∩ Ω ¯ j = ∅ if i ̸= j. such that Ω 0 0 Fig. 4 shows a typical example of this situation with q = 4. According to [16, Th. 9.1], each of the principal eigenvalues Σ0j (λ) := σ[L − λm, D, Ω0j ], j ∈ {1, . . . , q}, is a real analytic function of λ. Thus, Σ0 (λ) := min Σ0j (λ), 1≤j≤q
λ ∈ R,
(4.1)
is continuous. Thanks to the properties of the principal eigenvalues collected in [16, Ch. 8], the next result of a technical nature holds. Lemma 4.1.
Suppose a−1 (0) ⊂ Ω and the condition (A) holds. Then, the following assertions are true:
(a) Σ (λ) < Σ0 (λ) for all λ ∈ R. ¯j ⊂ Ω for all j ∈ {1, . . . , q}. Then, (b) Let Oj be an open subset of Ω of class C 2 such that Ω0j ⊂ Oj and O for every λ ∈ R, min σ[L − λm, D, Oj ] ≤ Σ0 (λ). (4.2) 1≤j≤q
(c) For sufficiently small δ > 0, set { } Ωδj ≡ Ω0j + Bδ (0) = x ∈ Ω : dist (x, Ω0j ) < δ ,
1 ≤ j ≤ q.
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13
Then, lim min σ[L − λm, D, Ωδj ] = Σ0 (λ) δ↓0 1≤j≤q
for all λ ∈ R.
(4.3)
Proof . By [16, Pr. 8.2], for every j ∈ {1, . . . , q} and λ ∈ R, Σ (λ) ≡ σ[L − λm, B, Ω ] < σ[L − λm, D, Ω0j ] ≡ Σ0j (λ). Thus, Σ (λ) < min Σ0j (λ) ≡ Σ0 (λ), 1≤j≤q
which concludes the proof of Part (a). Similarly, within the setting of Part (b), for every j ∈ {1, . . . , q} and λ ∈ R, σ[L − λm, D, Oj ] ≤ σ[L − λm, D, Ω0j ] and hence min σ[L − λm, D, Oj ] ≤ σ[L − λm, D, Ω0j ]
1≤j≤q
for all λ ∈ R and j ∈ {1, . . . , q}. Therefore, taking minima in j ∈ {1, . . . , q}, (4.2) holds for all λ ∈ R, which ends the proof of Part (b). Finally, thanks to [16, Th. 8.5], we have that, for every j ∈ {1, . . . , q} and λ ∈ R, lim σ[L − λm, D, Ωδj ] = σ[L − λm, D, Ω0j ]. δ↓0
Thus, taking minima in j ∈ {1, . . . , q}, (4.3) holds and the proof of the lemma is complete.
□
The next result theorem establishes that, under condition (A), the unique positive solutions of (1.4) are those of C+ ∪ C− . Theorem 4.1.
Suppose a−1 (0) ⊂ Ω and the condition (A) holds. Then:
(a) The problem (1.4) admits a positive solution if, and only if, Σ (λ) < 0 and Σ0 (λ) > 0.
(4.4)
(b) The condition (4.4) holds if and only if λ ∈ (λ∗− , λ− ) ∪ (λ+ , λ∗+ ). (c) The set of positive solutions of (1.4) consists of C− ∪ C+ \ {(λ± , 0)}. Proof . Let (λ, w) be a positive solution of (1.4). By Lemma 2.1, Σ (λ) < 0. Moreover, since a = 0 in Ω0j for all j ∈ {1, . . . , q}, it follows from (2.3) and [16, Prop. 8.2] that 0 = σ[L − λm + ah(w), B, Ω ] < σ[L − λm + ah(w), D, Ω0j ] = Σ0j (λ). Thus, Σ0 (λ) > 0. Therefore, (4.4) is necessary for the existence of a positive solution. Subsequently, we assume (4.4). The existence of arbitrarily small positive subsolutions can be accomplished as in the proof Theorem 3.1. Indeed, if φ is any positive eigenfunction associated to (L − λm, B, Ω ), then, the function w := εφ is a positive subsolution for (1.4) for sufficiently small ε > 0. To show the existence of arbitrarily large supersolutions under condition (4.4) we proceed as follows. ¯ ) and γ > 0 such that According to [16, Lem. 2.1], there exist ψ ∈ C 2 (Ω ∂ν ψ(x) ≥ γ
for all x ∈ Γ1 .
(4.5)
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Moreover, by Lemma 4.1(c), there exists δ > 0 such that 0 < min σ[L − λm, D, Ωδj ] < Σ0 (λ). 1≤j≤q
(4.6)
Note that, since a−1 (0) ⊂ Ω and (A) holds, δ > 0 can be shortened, if necessary, so that ¯ j ∩ ∂Ω = ∅ and Ω ¯i ∩ Ω ¯ j = ∅ for all i, j ∈ {1, . . . , q} with i ̸= j. Ω δ δ δ
(4.7)
For every j ∈ {1, . . . , q}, let denote by φjδ ≫ 0 the principal eigenfunction associated to σ[L − λm, D, Ωδj ] ¯ → R defined by normalized so that ∥φjδ ∥C(Ω) ¯ = 1, and consider the function ϕ : Ω ⎧ j j if x ∈ Ωδ/2 , j ∈ {1, . . . , q}, ⎪ ⎨ φδ (x) ⋃q M ψ(x) ¯ ϕ(x) := e if x ∈ Ω \ j=1 Ωδj , ⎪ j ⎩ υ (x) if x ∈ Ωδj \ Ωδ/2 , j ∈ {1, . . . , q}, j for some positive constant M > 0 and any positive smooth extension, υj (x), of the functions φjδ and eM ψ(x) j to Ωδj \ Ωδ/2 , j ∈ {1, . . . , q}. We claim that the function w := Cϕ is a positive supersolution of (1.4) for sufficiently large M > 0 and C > 0. Indeed, w ≥ 0 on Γ0 by construction. Moreover, by (4.5), it is apparent that B(w) = CeM ψ (M ∂ν ψ + β) ≥ CeM ψ (M γ + β) > 0 on Γ1 , for sufficiently large M > 0. Thus, Bw ≥ 0
on ∂Ω ,
for every C > 0 and sufficiently large M > 0. On the other hand, by (4.6), we have that, for every j ∈ {1, . . . , q}, j (L − λm)w = Cσ[L − λm, D, Ωδj ]φjδ > 0 ≥ −ah(w)w in Ωδ/2 for all C > 0. Finally, we have that (L − λm + ah(Cϕ))Cϕ ≥ 0
j in Ω \ ∪qj=1 Ωδ/2
provided − (L − λm)eM ψ ≤ ah(CeM ψ )eM ψ
¯ \ ∪q Ω j in Ω j=1 δ
(4.8)
and − (L − λm)υj ≤ ah(Cυj )υj
j in Ωδj \ Ωδ/2 for each j ∈ {1, . . . , q}.
(4.9)
As (4.8) and (4.9) hold true for sufficiently large C > 0 because, thanks to the assumption (A), there exists a constant µ > 0 such that aL :=
⋃qinf
Ω\
j=1
Ω
j δ/2
j a > 0 and ah(Cϕ) ≥ aL h(Cµ) in Ω \ ∪qj=1 Ωδ/2 ,
the proof of the previous claim is complete by taking into account that lim h(Cµ) = ∞.
C→∞
Lastly, shortening ε > 0, if necessary, we can suppose that w ≤ w. Therefore, according to Theorem 6.1, (1.4) admits a positive solution in [w, w], which concludes the proof of Part (a). As, according to Theorem 3.2, (1.4) has a positive solution for every λ ∈ J := (λ∗− , λ− ) ∪ (λ+ , λ∗+ ),
D. Aleja, I. Antón and J. López-Gómez / Nonlinear Analysis 192 (2020) 111690
15
Fig. 5. A genuine configuration satisfying assumption (B).
by Part (a) it becomes apparent that Σ (λ) < 0 and Σ0 (λ) > 0 for all λ ∈ J. Suppose λ∗+ ∈ R. Then, Σ (λ∗+ ) < 0 and Σ0 (λ∗+ ) ≥ 0. Suppose that Σ0 (λ∗+ ) > 0. Then, by the proof of Part (a), (1.4) admits a positive supersolution, w, and enlarging C > 0, if necessary, it is easily seen that w also is a positive supersolution of (1.4) for all λ ∈ (λ∗+ − η, λ∗+ + η) provided η > 0 is sufficiently small. Consequently, taking into account Theorem 3.1, for every λ ∈ (λ∗+ − η, λ∗+ + η) the problem (1.4) possesses a unique positive solution, w(λ), such that w(λ) ≤ w. As this contradicts (3.3), it becomes apparent that Σ0 (λ∗+ ) = 0. Similarly, Σ0 (λ∗− ) = 0 if λ∗− ∈ R. By the concavity properties of Σ (λ) and Σ0 (λ) established in [16, Ch. 9], the proof of Part (b) is complete. Part (c) is a byproduct of Parts (a) and (b) taking into account that, for every λ ∈ (λ∗− , λ− ) ∪ (λ+ , λ∗+ ), (1.4) has a positive solution in C− ∪ C+ and that the positive solution is unique. □ 5. Under assumption (B) the solution set consists of C± \ {(λ± , 0)} Throughout this section, instead of (A), we will assume that (B) a−1 (0) ⊂ K− ∪ K+ , where K− and K+ are two disjoint compact subsets of Ω such that m(x) < 0 for all x ∈ K− and m(x) > 0 for all x ∈ K+ . Under this assumption, we are not requiring any special structure, or regularity, for int a−1 (0) like in (A). Actually, the set of zeros of a can be general, as soon as it can be separated into two pieces, one in m−1 ((0, ∞)) and another one in m−1 ((−∞, 0)), as illustrated in Fig. 5, where the set of zeros of a consists of a spiraling family of tangent disks whose diameters approximate zero, in the region m > 0, plus a fuzzy set of small disks and points, in the region m < 0. In Fig. 5 the set m−1 (0) consists of a separating straight line separating m−1 ((0, ∞)) and m−1 ((−∞, 0)). The main result of this section can be stated as follows. Theorem 5.1. (a) (b) (c) (d)
The The The The
Suppose a−1 (0) ⊂ Ω and the condition (B) is satisfied. Then:
˜ if it admits it at some λ ˜ > λ+ . problem (1.4) possesses a positive solution for each λ ∈ (λ+ , λ] ˜ ˜ < λ− . problem (1.4) possesses a positive solution for each λ ∈ [λ, λ− ) if it admits it at some λ ∗ ∗ problem (1.4) has a positive solution if and only if λ ∈ (λ− , λ− ) ∪ (λ+ , λ+ ). set of positive solutions of (1.4) consists of C− ∪ C+ \ {(λ± , 0)}.
16
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˜ w) ˜ > λ+ and, for sufficiently small Proof . Suppose that (1.4) admits a positive solution (λ, ˜ for some λ δ > 0, set δ δ ¯δ (0), ¯δ (0). K− := K− + B K+ := K+ + B As K− and K+ are disjoint compact subsets of Ω , they are at a positive distance, as well as K± from ∂Ω . Hence, δ can be shortened, if necessary, so that { δ > 0 if x ∈ K+ , δ δ δ δ K− ∩ K+ = ∅, K+ ∪ K− ⊂ Ω , m(x) δ < 0 if x ∈ K− . Now, given any principal eigenfunction, φ+ through ⎧ ⎪ ˜ ⎪ w(x) ⎪ ⎨ φ+ (x) ϕ(x) := ⎪ eM ψ(x) ⎪ ⎪ ⎩ υ(x)
≫ 0, associated to λ+ , we consider the function ϕ defined if if if if
δ/2
x ∈ K+ , δ/2 x ∈ K− (, ) δ δ ¯ \ K− x∈Ω ∪ K+ , δ/2 δ/2 δ δ \ K+ ), x ∈ (K− \ K− ) ∪ (K+
where ψ(x) is the function defined in (4.5) and υ(x) is any positive smooth extension of w ˜ and eM ψ to δ/2 δ/2 δ Mψ δ K+ \ K+ , and of φ+ and e to K− \ K− . Next, we will show that, for sufficiently large C > 0, the ˜ Indeed, arguing as in function w := Cϕ provides us with a positive supersolution of (1.4) for all λ ∈ (λ+ , λ]. the proof of Theorem 4.1, it becomes apparent that Bw = CBeM ψ ≥ 0
on ∂Ω
δ/2 ˜ for sufficiently large M > 0. Moreover, since m(x) > 0 for all x ∈ K+ , we have that, for every λ ∈ (λ+ , λ] and C ≥ 1, δ/2 ˜ + ah(w)) (L − λm + ah(C w)) ˜ w ˜ ≥ (L − λm ˜ w ˜ = 0 in K+ . δ/2 ˜ and C > 0, Similarly, since m(x) < 0 for all x ∈ K , for every λ ∈ (λ+ , λ] −
(L − λm + ah(Cφ+ ))φ+ ≥ (L − λ+ m)φ+ = 0
δ/2
in K− .
Thus, for every C ≥ 1, we find that (L − λm + ah(w))w ≥ 0
δ/2
δ/2
in K− ∪ K+ .
Finally, since, by construction, ϕL :=
inf
ϕ > 0,
δ/2 δ/2 ¯ ∪K+ ) Ω\(K −
aL :=
inf
a > 0,
δ/2 δ/2 ¯ ∪K+ ) Ω\(K −
¯ \ (K δ/2 ∪ K δ/2 ), it becomes apparent that, in Ω + − (L − λm + ah(Cϕ))ϕ ≥ Lϕ − λmϕ + aL h(Cϕ)ϕ ≥ 0 ˜ for sufficiently large C > 0 and every λ ∈ (λ+ , λ]. Since, arguing as in the proof of Theorem 3.1, (1.4) possesses arbitrarily small positive subsolution, w := εφ, satisfying (6.1), by Theorem 6.1, (1.4) admits a positive solution in [w, w]. In particular, w(λ) ≤ w
˜ for all λ ∈ (λ+ , λ],
as illustrated in Fig. 6. This ends the proof of Part (a). The previous argument can be easily adapted to prove Part (b). By repetitive, we omit the details of the proof here in. According to Theorem 3.2, we already know that (1.4) admits a (unique) positive solution for every ˜ ≥ λ∗ . Then, by λ ∈ (λ∗− , λ− ) ∪ (λ+ , λ∗+ ). Suppose λ∗+ < ∞ and (1.4) has a positive solution for some λ + ˜ Since this contradicts (3.3), (1.4) the proof of Part (a), the curve (λ, w(λ)) must be bounded in λ ∈ (λ+ , λ]. cannot admit a positive solution if λ ≥ λ∗+ . Similarly, one can prove that (1.4) cannot admit a positive solution if λ∗− > −∞ and λ ≤ λ∗− . This completes the proof of Part (c). Part (d) is a byproduct of Part (c) and of Theorem 3.2. □
D. Aleja, I. Antón and J. López-Gómez / Nonlinear Analysis 192 (2020) 111690
17
Fig. 6. The unique positive solution is bounded above by w.
Appendix The main result of this section is the next refinement of a celebrated result of Amann [4]. It should be noted that here we are not requiring h(w) to be H¨older continuous in w, as in [4], though we are obligated to restrict the usual concept of subsolution. Theorem 6.1. Under the general assumptions of Section 1, suppose that w and w are a subsolution and a supersolution of (1.4), respectively, such that, for some α > 0, (L − λm + ah(w))w ≤ −αw
in Ω ,
(6.1)
and 0 ≤ w ≤ w.
(6.2)
Then, (1.4) possesses a minimal solution, w∗ , in the interval [w, w], i.e., w ≤ w∗ ≤ w ≤ w
(6.3)
w ≤ w ≤ w.
(6.4)
for every solution w of (1.4) such that
Note that in the proof of Theorem 3.1, w := εφ satisfies (6.1) with −α := Σ (λ) + ∥a∥C(Ω) ¯ h(ε) < 0. Proof . By the Stone–Weierstrass theorem, there exists a smooth function, hα : [0, M ] → R,
M ≡ ∥w∥C(Ω) ¯ ,
such that hα ≤ h ≤ hα + α/∥a∥C(Ω) ¯ in [0, M ].
(6.5)
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D. Aleja, I. Antón and J. López-Gómez / Nonlinear Analysis 192 (2020) 111690
Subsequently, for every α ≥ 0, we set gα (x, w) := λm(x)w − a(x)hα (w)w − αw,
h0 ≡ h,
and, for any fixed κ > −σ[L, B, Ω ], we consider the iterative schemes wn+1 := (L + κ)−1 (gα (·, wn ) + κwn ), wn+1 := (L + κ)
−1
w0 ≡ w,
(6.6)
w0 ≡ w.
(gα (·, wn ) + κwn ),
We claim that, in Ω , enlarging κ, if necessary, one has that gα (·, w) + κw ≤ gα (·, v) + κv
if
0 ≤ w ≤ v ≤ M.
(6.7)
Indeed, let µ > 0 denote the Lipschitz constant of hα (w)w in [0, M ]. Then, hα (v)v − hα (w)w ≤ µ(v − w)
if 0 ≤ w ≤ v ≤ M
and hence, κ(v − w) + gα (·, v) − gα (·, w) ≥ (κ + λm − α − aµ) (v − w) ≥ 0 provided κ is sufficiently large. For any of those κ’s, (6.7) holds. Next, we will show that w0 ≤ w1 and w1 ≤ w0 in Ω .
(6.8)
Indeed, since w satisfies (6.1) and w is a supersolution of (1.4), it follows from (6.5) that Lw ≤ gα (·, w)
and Lw ≥ gα (·, w)
in Ω .
Thus, (L + κ)(w1 − w) = gα (·, w) + κw − (L + κ)w = gα (·, w) − Lw ≥ 0. Moreover, B(w1 − w) = −Bw ≥ 0. Therefore, it follows from [16, Th. 7.10] that w1 −w ≥ 0. Similarly, w1 −w ≤ 0. So, (6.8) holds. Consequently, since (L + κ)−1 is strongly positive, it follows from (6.2), (6.7) and (6.8) that wn ≤ wn+1 ≤ wn+1 ≤ wn
for all n ≥ 0.
(6.9)
Thanks to (6.9), the sequences of continuous functions defined by H n := gα (·, wn ) + κwn
and H n := gα (·, wn ) + κwn ,
n ≥ 0,
¯ ). Thus, combining the compactness of (L + κ)−1 with (6.9), it is easily seen that there are bounded in C(Ω α α ¯ exist w , w ∈ C(Ω ) such that wα := lim wn n→∞
and w α := lim wn n→∞
¯ ). in C(Ω
Moreover, letting n → ∞ in (6.6), it is apparent that wα and w α are solutions of { Lw = (λm − ahα (w) − α) w in Ω , Bw = 0 on ∂Ω ,
(6.10)
(6.11)
such that w ≤ wα ≤ wα ≤ w.
(6.12)
D. Aleja, I. Antón and J. López-Gómez / Nonlinear Analysis 192 (2020) 111690
19
Next, choose a sequence {αi }i≥1 such that 0 < αi+1 < αi ≤ α,
lim αi = 0.
i→∞
As for every i ≥ 1 we have that (L − λm + ah(w))w ≤ −αw ≤ −αi w ≤ −αi+1 w
in Ω ,
we can repeat the previous process for α := αi to show the existence of two sequences, wi := wαi and w i := w αi of solutions of (6.11) for α := αi . For every κ > −σ[L, B, Ω ], these solutions satisfy ( ) ( ) wi = (L + κ)−1 gαi (·, wi ) + κwi and w i = (L + κ)−1 gαi (·, w i ) + κw i , (6.13) and, owing to (6.12), 0 ≤ w ≤ wi ≤ w i ≤ w ≤ M.
(6.14)
Thus, the functions G i := gαi (·, wi ) + κwi ,
Gi := gαi (·, w i ) + κw i ,
i ≥ 1,
¯ ). Note that, thanks to (6.5), are bounded in C(Ω ∥hαi ∥C[0,M ] ≤ ∥hαi − h∥C[0,M ] + ∥h∥C[0,M ] ≤ α/∥a∥C[0,M ] + ∥h∥C[0,M ] . ¯ ) such that along some subsequences of wi Thus, by the compactness of (L + κ)−1 , there exist w∗ , w∗ ∈ C(Ω i and w , relabeled by i, we have that w∗ := lim wi i→∞
and w∗ := lim w i i→∞
¯ ). in C(Ω
(6.15)
Letting i → ∞ in (6.13), it is apparent that w∗ and w∗ are solutions of (1.4). Moreover, by (6.14) and (6.15), they satisfy (6.16) 0 ≤ w ≤ w∗ ≤ w∗ ≤ w ≤ M. Finally, assume that w is a solution of (1.4) such that (6.4) holds. Then, repeating the previous process with w ≡ w as the supersolution, we can infer that 0 ≤ w ≤ w∗ ≤ w ≤ w ≤ M. The proof is complete.
□
References [1] D. Aleja, J. L´ opez-G´ omez, Concentration through large advection, J. Differential Equations 257 (2014) 3135–3164. [2] D. Aleja, J. L´ opez-G´ omez, Some paradoxical effects of the advection on a class of diffusive equations in ecology, Discrete Contin. Dyn. Syst. Ser. B 19 (2014) 3031–3056. [3] D. Aleja, J. L´ opez-G´ omez, Dynamics of a class of advective-diffusive equations in ecology, Adv. Nonlinear Stud. 15 (2015) 557–585. [4] H. Amann, Existence of multiple solutions for nonlinear elliptic boundary value problems, Indiana Univ. Math. J. 21 (1971–72) 925–935. [5] S. Cano-Casanova, J. L´ opez-G´ omez, Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems, J. Differential Equations 178 (2002) 123–211. [6] J.A. Cid, On extremal solutions in Schauder’s theorem with applications to differential equations, Bull. Belg. Math. Soc. 11 (2004) 15–20. [7] M.G. Crandall, P.H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971) 321–340. [8] D. Daners, J. L´ opez-G´ omez, Global dynamics of generalized logistic equations, Adv. Nonlinear Stud. 18 (2018) 217–236. [9] C. De Coster, P. Omari, Unstable periodic solutions for a parabolic problem in presence of non-well-ordered lower and upper solutions, J. Funct. Anal. 175 (2000) 52–88.
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[10] J. Esquinas, J. L´ opez-G´ omez, Optimal multiplicity in local bifurcation theory, I. Generalized generic eigenvalues, J. Differential Equations 71 (1988) 72–92. [11] S. Fern´ andez-Rinc´ on, J. L´ opez-G´ omez, The singular perturbation problem for a class of generalized logistic equations under non-classical mixed boundary conditions, Adv. Nonlinear Stud. 19 (2019) 1–27. [12] J.M. Fraile, P. Koch-Medina, J. L´ opez-G´ omez, S. Merino, Eigenvalue elliptic problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Differential Equations 127 (1996) 295–319. [13] Y. Li, L. Nirenberg, Regularity of the distance function to the boundary, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 29 (2005) 257–269. [14] J. L´ opez-G´ omez, The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems, J. Differential Equations 127 (1996) 263–294. [15] J. L´ opez-G´ omez, Spectral theory and nonlinear functional analysis, in: Chapman and Hall/CRC Research Notes in Mathematics, vol. 426, 2001, Boca Raton. [16] J. L´ opez-G´ omez, Linear Second Order Elliptic Operators, World Scientific Publishing, Singapore, 2013. [17] J. L´ opez-G´ omez, Metasolutions of Parabolic Problems in Population Dynamics, CRC Press, Boca Raton, 2015. [18] J. Murray, Mathematical Biology: I. An Introduction, Interdisciplinary Applied Mathematics, third ed., Springer-Verlag, New York, 2002.
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Nonlinear Analysis www.elsevier.com/locate/na
Solution components in a degenerate weighted BVP✩ D. Aleja a , I. Antón b , J. López-Gómez c ,∗ a
Departamento de Matemática Aplicada, Ciencia e Ingeniería de los Materiales y Tecnología Electrónica, Universidad Rey Juan Carlos, Móstoles (Madrid) 28933, Spain b Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, Madrid 28040, Spain c Instituto de Matemática Interdisciplinar (IMI), Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, Madrid 28040, Spain
article
info
Article history: Received 22 July 2019 Accepted 4 November 2019 Communicated by Vicentiu D. Radulescu MSC: primary 35K57 35B09 35J25
abstract This paper ascertains the structure of the set of positive solutions of a degenerate weighted logistic equation with a continuous kinetic under non-classical mixed boundary conditions. After establishing the uniqueness of the positive solution and the existence of two global components of positive solutions, it is shown that any component must be a continuous curve. Further, some general sufficient conditions are provided, in terms of the weight functions involved in the setting of the model, so that it possesses exactly two components of positive solutions. The problem of finding out the total number of components in the general case remains open. © 2019 Elsevier Ltd. All rights reserved.
Keywords: Generalized logistic equation Weighted problem Degenerate equation Non-classical mixed boundary conditions Structure of the positive solutions Counting the number of components
1. Introduction This paper studies the positive steady-states of the semilinear parabolic problem ⎧ in Ω × (0, ∞), ⎨ ∂t u + Lu = λm(x)u − a(x)h(u)u Bu = 0 on ∂Ω × (0, ∞), ⎩ u(·, 0) = u0 in Ω ,
(1.1)
where Ω is a bounded subdomain (open and connected set) of class C 2 of RN , N ≥ 1, L is a second order uniformly elliptic operator L := −div (A∇·) + ⟨b, ∇·⟩ + c, (1.2) ✩ This paper has been supported by the IMI of Complutense University and the Ministry of Education, Science and Universities of Spain under Research Grant PGC2018-097104-B-100. ∗ Corresponding author.
E-mail addresses: [email protected] (D. Aleja), [email protected] (I. Antón), [email protected] (J. López-Gómez). https://doi.org/10.1016/j.na.2019.111690 0362-546X/© 2019 Elsevier Ltd. All rights reserved.
D. Aleja, I. Antón and J. López-Gómez / Nonlinear Analysis 192 (2020) 111690
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with 1,∞ A ∈ Msym (Ω )), N (W
N
b ∈ (L∞ (Ω )) ,
and B is a non-classical mixed boundary operator { ξ on Γ0 , Bξ := ∂ν ξ + β(x)ξ on Γ1 ,
c ∈ L∞ (Ω ),
ξ ∈ C(Γ0 ) ⊕ C 1 (Ω ∪ Γ1 ),
(1.3)
where Γ0 and Γ1 are two disjoint open and closed subsets of ∂Ω such that ∂Ω := Γ0 ∪ Γ1 , β ∈ C 1 (Γ1 ), and ν := An is the co-normal vector field; n standing for the outward normal vector-field to Ω . Throughout this paper, for any pair of real functions h1 , h2 in Ω , it is said that h1 ⪈ h2 if h1 ≥ h2 but h1 ̸= h2 . The standard notation h1 > h2 might be slightly confusing, for as it could suggest that h1 (x) > h2 (x) for all x ∈ Ω . In ¯ ) is a continuous function that changes sign in the model (1.1), λ ∈ R is regarded as a parameter, m ∈ C(Ω ¯ ¯ Ω , a ∈ C(Ω ) satisfies a ⪈ 0 and the nonlinearity h ∈ C(R) is assumed to satisfy h(0) = 0,
h(u) > h(v) > 0 if u > v > 0, and
lim h(u) = ∞.
u→∞
As in applications to population dynamics the problem (1.1) models the evolution of a dispersing species, u, in the inhabiting territory Ω (see, e.g., Aleja & L´ opez-G´omez [1–3], as well as the monographs of Murray [18] ∞ and L´ opez-G´ omez [17]), the initial data, u0 ∈ L (Ω ), is assumed to satisfy u0 ⪈ 0. Under these assumptions, (1.1) possesses a unique classical solution, u(t, x; u0 ), which is strongly positive for all t > 0, and the dynamics of (1.1) is governed either by the non-negative solutions of the semilinear elliptic boundary value problem { Lw = λm(x)w − a(x)h(w)w in Ω , (1.4) Bw = 0 on ∂Ω , or by the underlying metasolutions of (1.4) (see, e.g., Fraile et al. [12] and Daners & L´opez-G´omez [8]). The main goal of this paper is ascertaining the global structure of the set of positive solutions of (1.4), (λ, w), w ⪈ 0, according to the nature of the weight functions m, a, and the size of the parameter λ ∈ R. This task is accomplished in two particular cases when the principal eigenvalue of the term (L − λm, B, Ω ), denoted throughout this paper by Σ (λ) ≡ σ[L − λm, B, Ω ],
λ ∈ R,
(1.5)
satisfies Σ (λ0 ) > 0 for some λ0 ∈ R. As it will be shown in Section 2, under this assumption, Σ −1 (0) = {λ− , λ+ } with λ− < λ0 < λ+ , and the set of solutions of (1.4) possesses two unbounded components, C− and C+ , bifurcating from zero at (λ− , 0) and (λ+ , 0), respectively, consisting of positive solutions of (1.4), except for the bifurcation points (λ± , 0). Besides generalizing very substantially all previous existence and uniqueness results up to cover the most general case when h(w) is merely continuous and β(x) changes of sign, so substantially generalizing the pioneering ones of J. M. Fraile et al. [12], this paper addresses the extremely challenging problem of ascertaining the exact number of components of the set of positive solutions of (1.4). Actually, our main aim is establishing that C+ and C− are the unique components of positive solutions of (1.4) under each of the following structural conditions on a−1 (0): (A) Ω0 ≡ int a−1 (0) ̸= ∅ is an open subset of class C 2 of Ω with finitely many components, Ω0j , 1 ≤ j ≤ q, ¯i ∩ Ω ¯ j = ∅ if i ̸= j. such that Ω 0 0 −1 (B) a (0) ⊂ K− ∪ K+ , where K− and K+ are two disjoint compact subsets of Ω such that m(x) < 0 for all x ∈ K− and m(x) > 0 for all x ∈ K+ . The condition (A) extends the condition of J. M. Fraile et al. [12] to cover the case when Ω0 consists of finitely many smooth components, while (B) does not impose any restriction on a−1 (0), except that m cannot vanish on it. From a technical point of view, in order to solve this problem, this paper contributes with the next three technical novelties, which might be useful in a variety of contexts:
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(i) In the regions where a(x) > 0, among them the edges of Ω , ∂Ω , the positive constants cannot provide us with supersolutions of (1.4) if β is somewhere negative. We can overcome this technical difficulty by using an exponential function, eM ψ , with sufficiently large M > 0, in order to absorb the negative ¯ ) is a smooth function with min∂Ω ∂ν ψ > 0. Such a part of β with the constant M . Here, ψ ∈ C 2 (Ω function exists by [16, Lem. 2.1]; its regularity equals the regularity of Ω (see Li & Nirenberg [13] and Fern´ andez-Rinc´ on & L´ opez-G´ omez [11] for any further required details). ˜ w), ˜ > λ+ (resp. λ ˜ < λ− ), must lie on (ii) The proof of the fact that any positive solution of (1.4), (λ, ˜ with λ C+ (resp. C− ) under condition (A), or (B), relies on the uniqueness of the positive solution established by Theorem 3.1 through the construction of a certain supersolution to prove that (1.4) has a positive ˜ (resp. λ ∈ [λ, ˜ λ− )). Each of the conditions (A) and (B) guarantees the solution for every λ ∈ (λ+ , λ] existence of such a supersolution. Although its construction is folklore when m has constant sign, it is far from obvious when m changes of sign. These constructions are extremely sharp because (1.4) cannot admit any positive solution if λ ∈ [λ− , λ+ ]. Both use the supersolutions eM ψ within the patches of Ω where a(x) > 0. (iii) This paper provides a (non-trivial) general method to prove that a component of the solution set must be a continuous curve if there is uniqueness. The method involves a tricky use of the uniqueness, the compactness of the associated resolvent operator, and the fact that the components are maximal closed and connected subsets. In the existing literature a global continuation method based on the implicit function theorem is commonly used to ascertain the structure of the solution set. As in this paper h(w) is far from smooth, the implicit function theorem cannot be invoked. This paper is organized as follows. Based on the global bifurcation results of [15], Section 2 establishes the existence of C− and C+ . Section 3 shows the uniqueness of the positive solution and infers from it that C+ and C− are continuous curves. Sections 4 and 5 show that these are the unique components of positive solutions under conditions (A) and (B), respectively. Finally, in Appendix we provide an abstract existence result that we could not find elsewhere. It is a refinement of a classical result of Amann [4]. Although the existence of monotone sequences starting from a couple of ordered sub and super solutions, u ≤ v, and converging respectively to the minimal and the maximal solutions of the problem in the interval [u, v] was proven without the restrictions of our Theorem 6.1 by C. De Coster and P. Omari in [9, Le. 2.7], we have preferred to deliver a self-contained proof of the precise result needed in this paper, Theorem 6.1, because [9] focused attention on the case of homogeneous Dirichlet boundary conditions, while here we are dealing with a non-classical mixed boundary operator. The existence of the minimal and the maximal solutions might be also established by adapting the topological argument of J. A. Cid [6]. 2. The two components bifurcating from (λ, 0) Throughout this paper, for any given term (L, B, Ω ), as those introduced in Section 1, σ[L, B, Ω ] stands for the principal eigenvalue of (L, B, Ω ), i.e., the unique value of τ for which the linear eigenvalue problem { Lφ = τ φ in Ω , Bφ = 0 on ∂Ω , admits a positive eigenfunction φ ⪈ 0. In such case, by e.g. [16, Ch. 7], φ ≫ 0, in the sense that φ(x) > 0 for all x ∈ Ω ∪ Γ1 and ∂ν φ(x) < 0 if x ∈ Γ0 . Also, we set Σ (λ) ≡ σ[L − λm, B, Ω ],
λ ∈ R.
(2.1)
4
D. Aleja, I. Antón and J. López-Gómez / Nonlinear Analysis 192 (2020) 111690
Fig. 1. The graph of Σ (λ).
This principal eigenvalue plays a pivotal role to characterize the existence of positive solutions of (1.4). According to either [16, Th. 9.1], or [14], Σ (λ) is analytic, strictly concave and, since m(x) changes of sign in Ω , lim Σ (λ) = −∞. λ→±∞
Therefore, if Σ (λ0 ) > 0 for some λ0 ∈ R, then ⎧ ⎨ <0 =0 Σ (λ) ⎩ >0
there exist λ− < λ0 < λ+ such that iff λ ∈ (−∞, λ− ) ∪ (λ+ , ∞), iff λ ∈ {λ− , λ+ }, iff λ ∈ (λ− , λ+ ).
(2.2)
Fig. 1 shows the graph of Σ (λ) in this case. The main goal of this section is establishing the existence of two components of the set of positive solutions of (1.4) emanating from the curve of trivial solutions (λ, 0) at λ = λ+ and λ = λ− . The next result establishes a necessary condition for the existence of a positive solution of (1.4). It reveals the importance of the sign of Σ (λ) in our analysis. Lemma 2.1.
If (1.4) admits a positive solution, w, then Σ (λ) < 0. Moreover, σ[L − λm + ah(w), B, Ω ] = 0
(2.3)
and w ≫ 0 in the sense that w(x) > 0 for all x ∈ Ω ∪ Γ1 and ∂ν w(x) < 0 if x ∈ Γ0 . Proof . Let w be a positive solution of (1.4). Then, w ⪈ 0 and { (L − λm + ah(w))w = 0 in Ω , Bw = 0 on ∂Ω , i.e., w is a positive eigenfunction associated with the eigenvalue zero of the term (L − λm + ah(w), B, Ω ). Thus, by the uniqueness of the principal eigenvalue, (2.3) holds and w ≫ 0. Consequently, owing to the monotonicity of the principal eigenvalue with respect to the potential established by S. Cano-Casanova and J. L´ opez-G´ omez in [5, Prop. 3.3], it is apparent that 0 = σ[L − λm + ah(w), B, Ω ] > σ[L − λm, B, Ω ] = Σ (λ), because ah(w) ⪈ 0. This ends the proof. □
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The next result establishes that the positive solutions of (1.4) can only bifurcate from (λ, 0) at the values of the parameter where Σ (λ) = 0. Lemma 2.2. Let λs ∈ R be such that (λ, w) = (λs , 0) is a bifurcation point to positive solutions of (1.4) from (λ, 0). Then, Σ (λs ) = 0. In particular, λs ∈ {λ− , λ+ } if Σ (λ0 ) > 0 for some λ0 ∈ R. Proof . By elliptic regularity, for every κ > −σ[L, B, Ω ], the positive solutions of (1.4) are the zeros (λ, w), ¯ ) → C(Ω ¯ ) defined by with w ⪈ 0, of the operator F : R × C(Ω F(λ, w) := w − (L + κ)−1 [(λm + κ − ah(w))w].
(2.4)
¯ ) such that wn ⪈ 0, Let (λn , wn ), n ≥ 1, be a sequence in R × C(Ω lim (λn , wn ) = (λs , 0) and F(λn , wn ) = 0 for all n ≥ 1.
n→∞
Then, dividing by ∥wn ∥C(Ω) ¯ , n ≥ 1, we have that [ ] wn wn −1 = (L + κ) (λn m + κ − ah(wn )) , ∥wn ∥C(Ω) ∥wn ∥C(Ω) ¯ ¯ Note that the sequence Hn := (λn m + κ − ah(wn ))
wn , ∥wn ∥C(Ω) ¯
n ≥ 1.
(2.5)
n ≥ 1,
¯ ), because is bounded in C(Ω ∥Hn ∥C(Ω) ¯ ≤ |λn |∥m∥C(Ω) ¯ + κ + ∥ah(wn )∥C(Ω) ¯ and lim
n→∞
(
) |λn |∥m∥C(Ω) = |λs |∥m∥C(Ω) ¯ + κ + ∥ah(wn )∥C(Ω) ¯ ¯ + κ.
¯ ) → C(Ω ¯ ), there exists a subsequence of Thus, by the compactness of the resolvent operator (L + κ)−1 : C(Ω wn /∥wn ∥C(Ω) ¯ , relabeled by n, such that lim
n→∞
wn =φ ∥wn ∥C(Ω) ¯
¯ ) for some φ ∈ C(Ω ¯ ). Necessarily, ∥φ∥ ¯ = 1 and φ ⪈ 0. Moreover, letting n → ∞ in (2.5) yields in C(Ω C(Ω) φ = (L + κ)−1 (λs m + κ)φ. By elliptic regularity, φ must be a strong solution of { (L − λs m)φ = 0 Bφ = 0
in Ω , on ∂Ω .
Consequently, since φ ⪈ 0, Σ (λs ) = 0 holds. This ends the proof.
□
The next result is the main theorem of this section. By a component it is meant a closed and connected subset that is maximal for the inclusion. Theorem 2.1. Suppose that Σ (λ0 ) > 0 for some λ0 ∈ R. Then, the set of non-negative solutions of (1.4) possesses two components, C+ and C− , such that (λ± , 0) ∈ C± , w ≫ 0 if (λ, w) ∈ C± \ {(λ± , 0)}, and C± ¯ ). Moreover, according to Lemma 2.1, are unbounded in R × C(Ω C− ⊂ {(λ, w) : λ < λ− , w ≫ 0} ∪ {(λ− , 0)}, C+ ⊂ {(λ, w) : λ > λ+ , w ≫ 0} ∪ {(λ+ , 0)}.
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Proof . Throughout this proof, for any fixed κ > −σ[L, B, Ω ], we consider the operators L(λ)w := w − (L + κ)−1 [(λm + κ)w],
N(w) := (L + κ)−1 (ah(w)w).
Since L(λ) is a compact perturbation of the identity map, it is a real analytic family of Fredholm operators of index zero. Moreover, N(w) ¯) lim = 0 in C(Ω w→0 ∥w∥C(Ω) ¯ and, due to (2.4), F(λ, w) = L(λ)w + N(w) satisfies the general structural assumptions of [15, Ch. 7]. By Lemma 2.2, λ± are the unique possible bifurcation values to positive solutions of (1.4) from (λ, 0). Moreover, owing to [16, Ch. 7], if φ± ≫ 0 stands for a principal eigenfunction associated to Σ (λ± ) = 0, then N [L(λ± )] = span {φ± } . On the other hand, since Σ (λ0 ) = σ[L − λ0 m, B, Ω ] > 0 and the principal eigenvalue is dominant, 0 cannot be an eigenvalue of (L − λ0 m, B, Ω ), which implies ¯ )). Therefore, according to [15, Sect. 4.4], the set of eigenvalues of L(λ), L(λ0 ) ∈ Iso(C(Ω Λ = {λ ∈ R : dim N [L(λ)] ≥ 1} , is discrete. Consequently, the algebraic multiplicity of Esquinas and L´opez-G´omez [10] of L(λ) at λ = λ± , χ[L(λ); λ± ], is well defined. Actually, χ[L(λ); λ+ ] = χ[L(λ); λ− ] = 1,
(2.6)
because λ− and λ+ are simple eigenvalues of (L(λ), L′ (λ)) as discussed by Crandall and Ravinowitz [7], i.e., L′ (λ± )φ± ∈ / R[L(λ± )]. Indeed, on the contrary, assume that L′ (λ± )φ± = −(L + κ)−1 (mφ± ) ∈ R [L(λ± )] . ¯ ) such that Then, there exists v± ∈ C(Ω v± = (L + κ)−1 (λ± mv± + κv± − mφ± ) or, equivalently, {
(L − λ± m)v± = −mφ± Bv = 0
in Ω , on ∂Ω .
But, thanks to [16, Th. 9.4], this is not possible because mφ± ∈ / R[L − λ± m]. Thus, (2.6) holds and, hence, due to [15, Th. 5.6.2], the Leray–Schauder degree of L(λ) in BR (0), R > 0, changes by 1 as λ crosses λ+ and λ− . Therefore, by [15, Th. 6.4.3 and Lem. 6.5.4], there exist two components of the set solutions of (2.5), C+ and C− , such that (λ± , 0) ∈ C± and w ≫ 0 if (λ, w) ∈ C± \ {(λ± , 0)}. Moreover, each of these components satisfies some of the next alternatives: ¯ ). (A1) C± is unbounded in R × C(Ω (A2) There exists µ± ∈ Λ \ {λ± } such that (µ± , 0) ∈ C± .
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7
(A3) C± contains a point (ρ± , y± ) ∈ R × (Y± \ {0}), where ∫ { } ¯ Y± := y ∈ C(Ω ) : yφ± = 0 . Ω
Note that ¯ ) = Y± ⊕ span {φ± } . C(Ω ¯ ). So, Y± is a supplement of N [L(λ± )] in C(Ω According to Lemma 2.2, the alternative (A2) cannot occur, since (λ± , 0) ∈ C± and, due to Lemma 2.1, (1.4) cannot admit a positive solution if λ ∈ (λ− , λ+ ). So, (λ∓ , 0) ∈ / C± . Alternative (A3) cannot occur neither, because the nontrivial solutions on Y± cannot be positive. Therefore, the alternative (A1) holds. The proof is complete. □ 3. The components C+ and C− are continuous curves The next result establishes the uniqueness of the positive solution of (1.4) if it exists. Theorem 3.1.
Suppose that (1.4) admits a positive solution for some λ ∈ R. Then, it is unique.
Proof . By Lemma 2.1, Σ (λ) < 0. Suppose that (1.4) admits two positive solutions, w ̸= v. Then, { (L − λm + ah(w))w = (L − λm + ah(v))v = 0 in Ω , Bw = Bv = 0 on ∂Ω .
(3.1)
Moreover, w ≫ 0, v ≫ 0 and σ[L − λm + ah(w), B, Ω ] = σ[L − λm + ah(v), B, Ω ] = 0. On the other hand, since h(Cw) ≥ h(w) for all C ≥ 1, it follows that (L − λm + ah(Cw))Cw ≥ C(L − λm + ah(w))w = 0, and hence, w := Cw provides us with a supersolution of (1.4) for all C ≥ 1. Now, let φ ≫ 0 be the unique principal eigenfunction associated to Σ (λ) normalized so that ∥φ∥C(Ω) ¯ = 1. Then, w := εφ provides us with a positive subsolution of (1.4) for sufficiently small ε > 0. Indeed, as for sufficiently small ε > 0 Σ (λ) ≤ Σ (λ) + ah(ε) < 0, it is apparent that (L − λm + ah(w))w = (Σ (λ) + ah(εφ))w ≤ (Σ (λ) + ah(ε))w ≤ 0. Thus, shortening ε > 0 and enlarging C > 1, if necessary, we can assume that 0 ≪ w = εφ ≤ min{w, v} ⪇ max{w, v} ≤ Cw = w, and hence, by Theorem 6.1 in the Appendix, which is a refinement of the main theorem of Amann [4], it becomes apparent that the minimal positive solution of (1.4) in the order interval [w, w], say w∗ , satisfies 0 ≪ w ≤ w∗ ≤ w
and
0 ≪ w ≤ w∗ ≤ v.
Therefore, without lost of generality, we can assume that v ⪇ w. Suppose that ah(v) = ah(w)
in Ω .
(3.2)
D. Aleja, I. Antón and J. López-Gómez / Nonlinear Analysis 192 (2020) 111690
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Then, 0 = (L − λm)(w − v) + a[h(w)w − h(v)v] = (L − λm + ah(v))(w − v). Thus, since w − v ⪈ 0 and σ[L − λm + ah(v), B, Ω ] = 0, by the strong positivity of the principal eigenfunction, w ≫ v. Hence, in the support of a(x), it is apparent that ah(v) < ah(w), which contradicts (3.2). Therefore, since v ⪇ w, ah(w) ⪈ ah(v)
in Ω .
So, 0 = σ[L − λm + ah(w), B, Ω ] > σ[L − λm + ah(v), B, Ω ] = 0, which is a contradiction. Consequently, (1.4) cannot admit two positive solutions. □ Through the rest of this paper, Pλ stands for the λ-projection operator, Pλ (λ, w) = λ. As Pλ is a continuous map and the components C± are connected, Pλ (C− ) and Pλ (C+ ) are connected in R. Thus, since (λ± , 0) ∈ C± and, owing to Lemma 2.1, Pλ (C± ) ∩ (λ− , λ+ ) = ∅, it becomes apparent that there exist λ∗− ∈ [−∞, λ− ) and λ∗+ ∈ (λ+ , ∞] such that { } Pλ (C− ) ∈ (λ∗− , λ− ], [λ∗− , λ− ]
{ } and Pλ (C+ ) ∈ [λ+ , λ∗+ ), [λ+ , λ∗+ ] .
The next result establishes that each of the components C+ and C− has the global structure of a continuous curve bifurcating from (λ, 0) at λ = λ+ and λ = λ− , respectively. So, both components are connected by arcs. Subsequently, for every λ ∈ Pλ (C± ) \ {λ± }, we will denote by w(λ) the unique positive solution of (1.4), and we set w(λ± ) := 0. Theorem 3.2.
¯ ), λ ↦→ w(λ), is continuous. Therefore, The map Pλ (C± ) → C(Ω C± = {(λ, w(λ)) : λ ∈ Pλ (C± )}
are continuous curves bifurcating from (λ, 0) at λ = λ± . Consequently, Pλ (C− ) = (λ∗− , λ− ],
Pλ (C+ ) = [λ+ , λ∗+ ),
and lim ∥w(λ)∥C(Ω) ¯ =∞
λ→λ∗ ±
if |λ∗± | < +∞.
(3.3)
D. Aleja, I. Antón and J. López-Gómez / Nonlinear Analysis 192 (2020) 111690
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Proof . To prove the continuity of w(λ) in λ it suffices to show that lim w(λn ) = w(λω )
n→∞
for any infinity sequence {λn }n≥1 of Pλ (C± ) such that lim λn = λω ∈ Pλ (C± ).
(3.4)
n→∞
As the proof is the same for both components, C± , we will detail it only for C+ . Our proof proceeds by contradiction. So, assume that there is a sequence {λn }n≥1 of Pλ (C+ ) such that (3.4) holds for some λω ∈ Pλ (C+ ) and lim ∥w(λn ) − w(λω )∥C(Ω) ¯ ̸= 0. n→∞
Then, there exist R > 0 and a subsequence of {λn }n≥1 , relabeled by n ≥ 1, such that ∥w(λn ) − w(λω )∥C(Ω) ¯ >R
for all n ≥ 1.
(3.5)
Suppose that w(λω ) is a positive solution of (1.4) and note that, according to Lemma 2.1, this entails that λω > λ+ . As λn , λω ∈ Pλ (C+ ) for all n ≥ 1, by the uniqueness of the positive solution established by Theorem 3.1, it follows from the definition of w(λ) that (λω , w(λω )), (λn , w(λn )) ∈ C+ for all n ≥ 1. ¯ ), it follows from (3.5) that, for every n ≥ 1, there exists µn ∈ Pλ (C+ ) Thus, since C+ is connected in R×C(Ω such that ∥w(µn ) − w(λω )∥C(Ω) (3.6) ¯ = R. We claim that µn can be chosen so that |µn − λω | ≤ |λn − λω |.
(3.7)
To prove this, we will argue by contradiction assuming that ∥w(µ) − w(λω )∥C(Ω) ¯ ̸= R
if µ ∈ Pλ (C+ ) satisfies |µ − λω | ≤ |λn − λω |.
(3.8)
Note that λn = ̸ λω for all n ≥ 1, as in the contrary case, we would contradict (3.5). Suppose that λn > λω , as illustrated in Fig. 2, and consider ˆ n, µ ˆn := inf M
ˆ n := {ρ ∈ Pλ (C+ ) ∩ (λn , +∞) : ∥w(ρ) − w(λω )∥ = R}. M
ˆ n ̸= ∅. Indeed, if M ˆ n = ∅, then, by (3.8), µn < λω and, since λω < λn , (1.4) would possess Note that M two positive solutions for λ = λω : one in the interior of the ball centered at w(λω ), itself, and another one ˆ n ̸= ∅ and µ in the exterior, which again contradicts Theorem 3.1. Thus, M ˆn is well defined. Moreover, by definition, µ ˆn ≥ λn . Next, we will show that actually µ ˆn > λn . The proof will proceed, again, by contradiction. Assume that ˆ n , ℓ ≥ 1, a sequence such that µ ˆn = λn for some n ≥ 1, and let ρℓ ∈ M lim ρℓ = µ ˆ n = λn ,
ℓ→∞
∥w(ρℓ ) − w(λω )∥ = R,
ℓ ≥ 1.
(3.9)
Thanks to (3.9), (ρℓ , w(ρℓ )), ℓ ≥ 1, provides us with a bounded sequence of positive solutions of (1.4) in ¯ ). Now, remember that the positive solutions of (1.4) are the zeros of the operator F defined in (2.4). C(Ω Thus, for every ℓ ≥ 1, w(ρℓ ) = (L + κ)−1 Hℓ where Hℓ := [ρℓ m + κ − ah(w(ρℓ ))]w(ρℓ ),
10
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Fig. 2. The construction of µ ˆn .
¯ ), it follows from the compactness of the operator (L + κ)−1 : C(Ω ¯ ) → C(Ω ¯) and, since Hℓ is bounded in C(Ω that there exists a subsequence of w(ρℓ ), labeled again by ℓ, such that lim w(ρℓ ) = w
ℓ→∞
¯ ). Moreover, letting ℓ → ∞ yields in C(Ω w(ρℓ ) = (L + κ)−1 Hℓ −→ w = (L + κ)−1 [(λn m + κ − ah(w))w]. Thus, by elliptic regularity, (λn , w) is a strong non-negative solution of (1.4). Moreover, w ̸= w(λn ) because ∥w − w(λω )∥C(Ω) ¯ = R,
∥w(λn ) − w(λω )∥C(Ω) ¯ > R,
which contradicts the uniqueness of the positive solution established by Theorem 3.1. Consequently, µ ˆn > λn . Therefore, for every λ ∈ (λn , µ ˆn ), (1.4) has, at least, two positive solutions: one in BR and another one in ¯) \ B ¯R , which again is impossible by Theorem 3.1. It should be noted that C+ cannot link (λω , w(λω )) C(Ω to (λn , w(λn )) through some (ρ, w(ρ)) with ρ < λω ,
∥w(ρ) − w(λω )∥C(Ω) ¯ = R,
because w(λω ) is the unique positive solution of (1.4) at λω . This argument can be easily adapted to get another contradiction in the case when λn < λω by considering ˜ n, µ ˜n := sup M
˜ n := {ρ ∈ Pλ (C+ ) ∩ (−∞, λn ) : ∥w(ρ) − w(λω )∥ = R}, M
but we refrain ourselves of giving any further details by repetitive. This contradiction shows that the µn ’s can be chosen to satisfy (3.7). According to (3.4) and (3.7), we can infer that lim µn = λω .
n→∞
(3.10)
¯ ). Moreover, thanks to (3.6), (µn , w(µn )), n ≥ 1, is a bounded sequence of positive solutions of (1.4) in C(Ω Consequently, adapting the previous compactness argument, along some subsequence, relabeled by n, it can be easily seen that lim w(µn ) = w n→∞
D. Aleja, I. Antón and J. López-Gómez / Nonlinear Analysis 192 (2020) 111690
11
¯ ), where (λω , w) is a positive solution of (1.4) with in C(Ω ∥w − w(λω )∥C(Ω) ¯ = R. Therefore, (1.4) possesses two positive solutions at λ = λω , which, once more, contradicts Theorem 3.1. This contradiction shows that λω = λ+ and w(λ+ ) = 0. But in such case, according to Lemma 2.1, (1.4) cannot admit a positive solution, which contradicts the fact that (λ+ , w) is a positive solution. Note that, in this case, by construction, ∥w∥C(Ω) ¯ =R and hence w must be positive. This contradiction ends the proof of the continuity of the map λ ↦→ w(λ) and shows that indeed C+ and C− are continuous curves bifurcating from (λ, 0) at λ = λ+ and λ = λ− , respectively. Now, suppose, e.g., that Pλ (C+ ) = [λ+ , λ∗+ ] for some λ+ < λ∗+ < +∞. Then, since w(λ) is continuous in λ, C+ = {(λ, w(λ)) : λ ∈ [λ+ , λ∗+ ]} is a compact arc of continuous curve. In particular, C+ must be bounded, which contradicts Theorem 2.1. Therefore, Pλ (C+ ) = [λ+ , λ∗+ ) for some λ+ < λ∗+ ≤ +∞. (3.11) Similarly, it is easily seen that Pλ (C− ) = (λ∗− , λ− ] for some − ∞ ≤ λ∗− < λ− .
(3.12)
Finally, suppose that λ∗+ < +∞ in (3.11). Then, since for sufficiently small ε > 0 the set {(λ, w(λ)) : λ ∈ [λ+ , λ∗+ − ε]} is bounded, because it is a compact arc of continuous curve, it becomes apparent that lim sup ∥w(λ)∥C(Ω) ¯ = +∞ λ↑λ∗ +
¯ ). Similarly, since C+ is unbounded in R × C(Ω lim sup ∥w(λ)∥C(Ω) ¯ = +∞ λ↓λ∗ −
if λ∗− > −∞. The proof of (3.3) can be easily accomplished with an additional compactness argument within the vain of the previous one. Indeed, if (3.3) does not hold at, e.g., λ∗+ , then there exists a sequence λn , n ≥ 1, ¯ ). Thus, such that limn→∞ λn = λ∗+ with (λn , w(λn )) ∈ C+ and such that w(λn ), n ≥ 1, is bounded in C(Ω thanks to the previous compactness argument, along some subsequence, relabeled by n, we have that lim w(λn ) = w∗ ≥ 0
n→∞
where (λ∗+ , w∗ ) is a non-negative solution of (1.4). Since (λ∗+ , w∗ ) ∈ C+ and λ∗+ > λ+ , we find that w∗ ≫ 0 and therefore, λ∗+ ∈ Pλ (C+ ), which contradicts the fact that Pλ (C+ ) = [λ+ , λ∗+ ). This concludes the proof. □ Fig. 3 shows two admissible components C− and C+ in the special case when λ∗− and λ∗+ are real numbers. Since m(x) changes sign, the mapping λ ↦→ w(λ) might not be monotone.
D. Aleja, I. Antón and J. López-Gómez / Nonlinear Analysis 192 (2020) 111690
12
Fig. 3. An admissible global bifurcation diagram.
Fig. 4. An admissible case for hypothesis (A).
4. Under assumption (A) the solution set consists of C± \ {(λ± , 0)} Throughout this section, we assume that (A) Ω0 ≡ int a−1 (0) ̸= ∅ is an open subset of class C 2 of Ω with finitely many components, Ω0j , 1 ≤ j ≤ q, ¯i ∩ Ω ¯ j = ∅ if i ̸= j. such that Ω 0 0 Fig. 4 shows a typical example of this situation with q = 4. According to [16, Th. 9.1], each of the principal eigenvalues Σ0j (λ) := σ[L − λm, D, Ω0j ], j ∈ {1, . . . , q}, is a real analytic function of λ. Thus, Σ0 (λ) := min Σ0j (λ), 1≤j≤q
λ ∈ R,
(4.1)
is continuous. Thanks to the properties of the principal eigenvalues collected in [16, Ch. 8], the next result of a technical nature holds. Lemma 4.1.
Suppose a−1 (0) ⊂ Ω and the condition (A) holds. Then, the following assertions are true:
(a) Σ (λ) < Σ0 (λ) for all λ ∈ R. ¯j ⊂ Ω for all j ∈ {1, . . . , q}. Then, (b) Let Oj be an open subset of Ω of class C 2 such that Ω0j ⊂ Oj and O for every λ ∈ R, min σ[L − λm, D, Oj ] ≤ Σ0 (λ). (4.2) 1≤j≤q
(c) For sufficiently small δ > 0, set { } Ωδj ≡ Ω0j + Bδ (0) = x ∈ Ω : dist (x, Ω0j ) < δ ,
1 ≤ j ≤ q.
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13
Then, lim min σ[L − λm, D, Ωδj ] = Σ0 (λ) δ↓0 1≤j≤q
for all λ ∈ R.
(4.3)
Proof . By [16, Pr. 8.2], for every j ∈ {1, . . . , q} and λ ∈ R, Σ (λ) ≡ σ[L − λm, B, Ω ] < σ[L − λm, D, Ω0j ] ≡ Σ0j (λ). Thus, Σ (λ) < min Σ0j (λ) ≡ Σ0 (λ), 1≤j≤q
which concludes the proof of Part (a). Similarly, within the setting of Part (b), for every j ∈ {1, . . . , q} and λ ∈ R, σ[L − λm, D, Oj ] ≤ σ[L − λm, D, Ω0j ] and hence min σ[L − λm, D, Oj ] ≤ σ[L − λm, D, Ω0j ]
1≤j≤q
for all λ ∈ R and j ∈ {1, . . . , q}. Therefore, taking minima in j ∈ {1, . . . , q}, (4.2) holds for all λ ∈ R, which ends the proof of Part (b). Finally, thanks to [16, Th. 8.5], we have that, for every j ∈ {1, . . . , q} and λ ∈ R, lim σ[L − λm, D, Ωδj ] = σ[L − λm, D, Ω0j ]. δ↓0
Thus, taking minima in j ∈ {1, . . . , q}, (4.3) holds and the proof of the lemma is complete.
□
The next result theorem establishes that, under condition (A), the unique positive solutions of (1.4) are those of C+ ∪ C− . Theorem 4.1.
Suppose a−1 (0) ⊂ Ω and the condition (A) holds. Then:
(a) The problem (1.4) admits a positive solution if, and only if, Σ (λ) < 0 and Σ0 (λ) > 0.
(4.4)
(b) The condition (4.4) holds if and only if λ ∈ (λ∗− , λ− ) ∪ (λ+ , λ∗+ ). (c) The set of positive solutions of (1.4) consists of C− ∪ C+ \ {(λ± , 0)}. Proof . Let (λ, w) be a positive solution of (1.4). By Lemma 2.1, Σ (λ) < 0. Moreover, since a = 0 in Ω0j for all j ∈ {1, . . . , q}, it follows from (2.3) and [16, Prop. 8.2] that 0 = σ[L − λm + ah(w), B, Ω ] < σ[L − λm + ah(w), D, Ω0j ] = Σ0j (λ). Thus, Σ0 (λ) > 0. Therefore, (4.4) is necessary for the existence of a positive solution. Subsequently, we assume (4.4). The existence of arbitrarily small positive subsolutions can be accomplished as in the proof Theorem 3.1. Indeed, if φ is any positive eigenfunction associated to (L − λm, B, Ω ), then, the function w := εφ is a positive subsolution for (1.4) for sufficiently small ε > 0. To show the existence of arbitrarily large supersolutions under condition (4.4) we proceed as follows. ¯ ) and γ > 0 such that According to [16, Lem. 2.1], there exist ψ ∈ C 2 (Ω ∂ν ψ(x) ≥ γ
for all x ∈ Γ1 .
(4.5)
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D. Aleja, I. Antón and J. López-Gómez / Nonlinear Analysis 192 (2020) 111690
Moreover, by Lemma 4.1(c), there exists δ > 0 such that 0 < min σ[L − λm, D, Ωδj ] < Σ0 (λ). 1≤j≤q
(4.6)
Note that, since a−1 (0) ⊂ Ω and (A) holds, δ > 0 can be shortened, if necessary, so that ¯ j ∩ ∂Ω = ∅ and Ω ¯i ∩ Ω ¯ j = ∅ for all i, j ∈ {1, . . . , q} with i ̸= j. Ω δ δ δ
(4.7)
For every j ∈ {1, . . . , q}, let denote by φjδ ≫ 0 the principal eigenfunction associated to σ[L − λm, D, Ωδj ] ¯ → R defined by normalized so that ∥φjδ ∥C(Ω) ¯ = 1, and consider the function ϕ : Ω ⎧ j j if x ∈ Ωδ/2 , j ∈ {1, . . . , q}, ⎪ ⎨ φδ (x) ⋃q M ψ(x) ¯ ϕ(x) := e if x ∈ Ω \ j=1 Ωδj , ⎪ j ⎩ υ (x) if x ∈ Ωδj \ Ωδ/2 , j ∈ {1, . . . , q}, j for some positive constant M > 0 and any positive smooth extension, υj (x), of the functions φjδ and eM ψ(x) j to Ωδj \ Ωδ/2 , j ∈ {1, . . . , q}. We claim that the function w := Cϕ is a positive supersolution of (1.4) for sufficiently large M > 0 and C > 0. Indeed, w ≥ 0 on Γ0 by construction. Moreover, by (4.5), it is apparent that B(w) = CeM ψ (M ∂ν ψ + β) ≥ CeM ψ (M γ + β) > 0 on Γ1 , for sufficiently large M > 0. Thus, Bw ≥ 0
on ∂Ω ,
for every C > 0 and sufficiently large M > 0. On the other hand, by (4.6), we have that, for every j ∈ {1, . . . , q}, j (L − λm)w = Cσ[L − λm, D, Ωδj ]φjδ > 0 ≥ −ah(w)w in Ωδ/2 for all C > 0. Finally, we have that (L − λm + ah(Cϕ))Cϕ ≥ 0
j in Ω \ ∪qj=1 Ωδ/2
provided − (L − λm)eM ψ ≤ ah(CeM ψ )eM ψ
¯ \ ∪q Ω j in Ω j=1 δ
(4.8)
and − (L − λm)υj ≤ ah(Cυj )υj
j in Ωδj \ Ωδ/2 for each j ∈ {1, . . . , q}.
(4.9)
As (4.8) and (4.9) hold true for sufficiently large C > 0 because, thanks to the assumption (A), there exists a constant µ > 0 such that aL :=
⋃qinf
Ω\
j=1
Ω
j δ/2
j a > 0 and ah(Cϕ) ≥ aL h(Cµ) in Ω \ ∪qj=1 Ωδ/2 ,
the proof of the previous claim is complete by taking into account that lim h(Cµ) = ∞.
C→∞
Lastly, shortening ε > 0, if necessary, we can suppose that w ≤ w. Therefore, according to Theorem 6.1, (1.4) admits a positive solution in [w, w], which concludes the proof of Part (a). As, according to Theorem 3.2, (1.4) has a positive solution for every λ ∈ J := (λ∗− , λ− ) ∪ (λ+ , λ∗+ ),
D. Aleja, I. Antón and J. López-Gómez / Nonlinear Analysis 192 (2020) 111690
15
Fig. 5. A genuine configuration satisfying assumption (B).
by Part (a) it becomes apparent that Σ (λ) < 0 and Σ0 (λ) > 0 for all λ ∈ J. Suppose λ∗+ ∈ R. Then, Σ (λ∗+ ) < 0 and Σ0 (λ∗+ ) ≥ 0. Suppose that Σ0 (λ∗+ ) > 0. Then, by the proof of Part (a), (1.4) admits a positive supersolution, w, and enlarging C > 0, if necessary, it is easily seen that w also is a positive supersolution of (1.4) for all λ ∈ (λ∗+ − η, λ∗+ + η) provided η > 0 is sufficiently small. Consequently, taking into account Theorem 3.1, for every λ ∈ (λ∗+ − η, λ∗+ + η) the problem (1.4) possesses a unique positive solution, w(λ), such that w(λ) ≤ w. As this contradicts (3.3), it becomes apparent that Σ0 (λ∗+ ) = 0. Similarly, Σ0 (λ∗− ) = 0 if λ∗− ∈ R. By the concavity properties of Σ (λ) and Σ0 (λ) established in [16, Ch. 9], the proof of Part (b) is complete. Part (c) is a byproduct of Parts (a) and (b) taking into account that, for every λ ∈ (λ∗− , λ− ) ∪ (λ+ , λ∗+ ), (1.4) has a positive solution in C− ∪ C+ and that the positive solution is unique. □ 5. Under assumption (B) the solution set consists of C± \ {(λ± , 0)} Throughout this section, instead of (A), we will assume that (B) a−1 (0) ⊂ K− ∪ K+ , where K− and K+ are two disjoint compact subsets of Ω such that m(x) < 0 for all x ∈ K− and m(x) > 0 for all x ∈ K+ . Under this assumption, we are not requiring any special structure, or regularity, for int a−1 (0) like in (A). Actually, the set of zeros of a can be general, as soon as it can be separated into two pieces, one in m−1 ((0, ∞)) and another one in m−1 ((−∞, 0)), as illustrated in Fig. 5, where the set of zeros of a consists of a spiraling family of tangent disks whose diameters approximate zero, in the region m > 0, plus a fuzzy set of small disks and points, in the region m < 0. In Fig. 5 the set m−1 (0) consists of a separating straight line separating m−1 ((0, ∞)) and m−1 ((−∞, 0)). The main result of this section can be stated as follows. Theorem 5.1. (a) (b) (c) (d)
The The The The
Suppose a−1 (0) ⊂ Ω and the condition (B) is satisfied. Then:
˜ if it admits it at some λ ˜ > λ+ . problem (1.4) possesses a positive solution for each λ ∈ (λ+ , λ] ˜ ˜ < λ− . problem (1.4) possesses a positive solution for each λ ∈ [λ, λ− ) if it admits it at some λ ∗ ∗ problem (1.4) has a positive solution if and only if λ ∈ (λ− , λ− ) ∪ (λ+ , λ+ ). set of positive solutions of (1.4) consists of C− ∪ C+ \ {(λ± , 0)}.
16
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˜ w) ˜ > λ+ and, for sufficiently small Proof . Suppose that (1.4) admits a positive solution (λ, ˜ for some λ δ > 0, set δ δ ¯δ (0), ¯δ (0). K− := K− + B K+ := K+ + B As K− and K+ are disjoint compact subsets of Ω , they are at a positive distance, as well as K± from ∂Ω . Hence, δ can be shortened, if necessary, so that { δ > 0 if x ∈ K+ , δ δ δ δ K− ∩ K+ = ∅, K+ ∪ K− ⊂ Ω , m(x) δ < 0 if x ∈ K− . Now, given any principal eigenfunction, φ+ through ⎧ ⎪ ˜ ⎪ w(x) ⎪ ⎨ φ+ (x) ϕ(x) := ⎪ eM ψ(x) ⎪ ⎪ ⎩ υ(x)
≫ 0, associated to λ+ , we consider the function ϕ defined if if if if
δ/2
x ∈ K+ , δ/2 x ∈ K− (, ) δ δ ¯ \ K− x∈Ω ∪ K+ , δ/2 δ/2 δ δ \ K+ ), x ∈ (K− \ K− ) ∪ (K+
where ψ(x) is the function defined in (4.5) and υ(x) is any positive smooth extension of w ˜ and eM ψ to δ/2 δ/2 δ Mψ δ K+ \ K+ , and of φ+ and e to K− \ K− . Next, we will show that, for sufficiently large C > 0, the ˜ Indeed, arguing as in function w := Cϕ provides us with a positive supersolution of (1.4) for all λ ∈ (λ+ , λ]. the proof of Theorem 4.1, it becomes apparent that Bw = CBeM ψ ≥ 0
on ∂Ω
δ/2 ˜ for sufficiently large M > 0. Moreover, since m(x) > 0 for all x ∈ K+ , we have that, for every λ ∈ (λ+ , λ] and C ≥ 1, δ/2 ˜ + ah(w)) (L − λm + ah(C w)) ˜ w ˜ ≥ (L − λm ˜ w ˜ = 0 in K+ . δ/2 ˜ and C > 0, Similarly, since m(x) < 0 for all x ∈ K , for every λ ∈ (λ+ , λ] −
(L − λm + ah(Cφ+ ))φ+ ≥ (L − λ+ m)φ+ = 0
δ/2
in K− .
Thus, for every C ≥ 1, we find that (L − λm + ah(w))w ≥ 0
δ/2
δ/2
in K− ∪ K+ .
Finally, since, by construction, ϕL :=
inf
ϕ > 0,
δ/2 δ/2 ¯ ∪K+ ) Ω\(K −
aL :=
inf
a > 0,
δ/2 δ/2 ¯ ∪K+ ) Ω\(K −
¯ \ (K δ/2 ∪ K δ/2 ), it becomes apparent that, in Ω + − (L − λm + ah(Cϕ))ϕ ≥ Lϕ − λmϕ + aL h(Cϕ)ϕ ≥ 0 ˜ for sufficiently large C > 0 and every λ ∈ (λ+ , λ]. Since, arguing as in the proof of Theorem 3.1, (1.4) possesses arbitrarily small positive subsolution, w := εφ, satisfying (6.1), by Theorem 6.1, (1.4) admits a positive solution in [w, w]. In particular, w(λ) ≤ w
˜ for all λ ∈ (λ+ , λ],
as illustrated in Fig. 6. This ends the proof of Part (a). The previous argument can be easily adapted to prove Part (b). By repetitive, we omit the details of the proof here in. According to Theorem 3.2, we already know that (1.4) admits a (unique) positive solution for every ˜ ≥ λ∗ . Then, by λ ∈ (λ∗− , λ− ) ∪ (λ+ , λ∗+ ). Suppose λ∗+ < ∞ and (1.4) has a positive solution for some λ + ˜ Since this contradicts (3.3), (1.4) the proof of Part (a), the curve (λ, w(λ)) must be bounded in λ ∈ (λ+ , λ]. cannot admit a positive solution if λ ≥ λ∗+ . Similarly, one can prove that (1.4) cannot admit a positive solution if λ∗− > −∞ and λ ≤ λ∗− . This completes the proof of Part (c). Part (d) is a byproduct of Part (c) and of Theorem 3.2. □
D. Aleja, I. Antón and J. López-Gómez / Nonlinear Analysis 192 (2020) 111690
17
Fig. 6. The unique positive solution is bounded above by w.
Appendix The main result of this section is the next refinement of a celebrated result of Amann [4]. It should be noted that here we are not requiring h(w) to be H¨older continuous in w, as in [4], though we are obligated to restrict the usual concept of subsolution. Theorem 6.1. Under the general assumptions of Section 1, suppose that w and w are a subsolution and a supersolution of (1.4), respectively, such that, for some α > 0, (L − λm + ah(w))w ≤ −αw
in Ω ,
(6.1)
and 0 ≤ w ≤ w.
(6.2)
Then, (1.4) possesses a minimal solution, w∗ , in the interval [w, w], i.e., w ≤ w∗ ≤ w ≤ w
(6.3)
w ≤ w ≤ w.
(6.4)
for every solution w of (1.4) such that
Note that in the proof of Theorem 3.1, w := εφ satisfies (6.1) with −α := Σ (λ) + ∥a∥C(Ω) ¯ h(ε) < 0. Proof . By the Stone–Weierstrass theorem, there exists a smooth function, hα : [0, M ] → R,
M ≡ ∥w∥C(Ω) ¯ ,
such that hα ≤ h ≤ hα + α/∥a∥C(Ω) ¯ in [0, M ].
(6.5)
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Subsequently, for every α ≥ 0, we set gα (x, w) := λm(x)w − a(x)hα (w)w − αw,
h0 ≡ h,
and, for any fixed κ > −σ[L, B, Ω ], we consider the iterative schemes wn+1 := (L + κ)−1 (gα (·, wn ) + κwn ), wn+1 := (L + κ)
−1
w0 ≡ w,
(6.6)
w0 ≡ w.
(gα (·, wn ) + κwn ),
We claim that, in Ω , enlarging κ, if necessary, one has that gα (·, w) + κw ≤ gα (·, v) + κv
if
0 ≤ w ≤ v ≤ M.
(6.7)
Indeed, let µ > 0 denote the Lipschitz constant of hα (w)w in [0, M ]. Then, hα (v)v − hα (w)w ≤ µ(v − w)
if 0 ≤ w ≤ v ≤ M
and hence, κ(v − w) + gα (·, v) − gα (·, w) ≥ (κ + λm − α − aµ) (v − w) ≥ 0 provided κ is sufficiently large. For any of those κ’s, (6.7) holds. Next, we will show that w0 ≤ w1 and w1 ≤ w0 in Ω .
(6.8)
Indeed, since w satisfies (6.1) and w is a supersolution of (1.4), it follows from (6.5) that Lw ≤ gα (·, w)
and Lw ≥ gα (·, w)
in Ω .
Thus, (L + κ)(w1 − w) = gα (·, w) + κw − (L + κ)w = gα (·, w) − Lw ≥ 0. Moreover, B(w1 − w) = −Bw ≥ 0. Therefore, it follows from [16, Th. 7.10] that w1 −w ≥ 0. Similarly, w1 −w ≤ 0. So, (6.8) holds. Consequently, since (L + κ)−1 is strongly positive, it follows from (6.2), (6.7) and (6.8) that wn ≤ wn+1 ≤ wn+1 ≤ wn
for all n ≥ 0.
(6.9)
Thanks to (6.9), the sequences of continuous functions defined by H n := gα (·, wn ) + κwn
and H n := gα (·, wn ) + κwn ,
n ≥ 0,
¯ ). Thus, combining the compactness of (L + κ)−1 with (6.9), it is easily seen that there are bounded in C(Ω α α ¯ exist w , w ∈ C(Ω ) such that wα := lim wn n→∞
and w α := lim wn n→∞
¯ ). in C(Ω
Moreover, letting n → ∞ in (6.6), it is apparent that wα and w α are solutions of { Lw = (λm − ahα (w) − α) w in Ω , Bw = 0 on ∂Ω ,
(6.10)
(6.11)
such that w ≤ wα ≤ wα ≤ w.
(6.12)
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19
Next, choose a sequence {αi }i≥1 such that 0 < αi+1 < αi ≤ α,
lim αi = 0.
i→∞
As for every i ≥ 1 we have that (L − λm + ah(w))w ≤ −αw ≤ −αi w ≤ −αi+1 w
in Ω ,
we can repeat the previous process for α := αi to show the existence of two sequences, wi := wαi and w i := w αi of solutions of (6.11) for α := αi . For every κ > −σ[L, B, Ω ], these solutions satisfy ( ) ( ) wi = (L + κ)−1 gαi (·, wi ) + κwi and w i = (L + κ)−1 gαi (·, w i ) + κw i , (6.13) and, owing to (6.12), 0 ≤ w ≤ wi ≤ w i ≤ w ≤ M.
(6.14)
Thus, the functions G i := gαi (·, wi ) + κwi ,
Gi := gαi (·, w i ) + κw i ,
i ≥ 1,
¯ ). Note that, thanks to (6.5), are bounded in C(Ω ∥hαi ∥C[0,M ] ≤ ∥hαi − h∥C[0,M ] + ∥h∥C[0,M ] ≤ α/∥a∥C[0,M ] + ∥h∥C[0,M ] . ¯ ) such that along some subsequences of wi Thus, by the compactness of (L + κ)−1 , there exist w∗ , w∗ ∈ C(Ω i and w , relabeled by i, we have that w∗ := lim wi i→∞
and w∗ := lim w i i→∞
¯ ). in C(Ω
(6.15)
Letting i → ∞ in (6.13), it is apparent that w∗ and w∗ are solutions of (1.4). Moreover, by (6.14) and (6.15), they satisfy (6.16) 0 ≤ w ≤ w∗ ≤ w∗ ≤ w ≤ M. Finally, assume that w is a solution of (1.4) such that (6.4) holds. Then, repeating the previous process with w ≡ w as the supersolution, we can infer that 0 ≤ w ≤ w∗ ≤ w ≤ w ≤ M. The proof is complete.
□
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