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Superlinear elliptic systems with reaction terms involving product of powers
Accepted Manuscript Superlinear elliptic systems with reaction terms involving product of powers
M. Chhetri, P. Girg
PII: DOI: Reference:
S0893-9659(17)30236-7 http://dx.doi.org/10.1016/j.aml.2017.07.005 AML 5299
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Applied Mathematics Letters
Received date : 22 May 2017 Revised date : 17 July 2017 Accepted date : 18 July 2017 Please cite this article as: M. Chhetri, P. Girg, Superlinear elliptic systems with reaction terms involving product of powers, Appl. Math. Lett. (2017), http://dx.doi.org/10.1016/j.aml.2017.07.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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SUPERLINEAR ELLIPTIC SYSTEMS WITH REACTION TERMS INVOLVING PRODUCT OF POWERS M. CHHETRI AND P. GIRG
Abstract. We consider a system of the form −∆u = λg1 (x, u, v)
where λ > 0 is a parameter, Ω ⊂
−∆v = λg2 (x, u, v)
RN (N
u=0=v
in
Ω;
in
Ω;
on
∂Ω,
≥ 2) is a bounded domain with sufficiently smooth boundary
∂Ω (a bounded open interval if N = 1). Here gi (x, s, t) : Ω × [0, +∞) × [0, +∞) → R (i = 1, 2) are
Carath´ eodory functions that exhibit superlinear growth at infinity involving product of powers of u and v. Using re-scaling argument combined with Leray-Schauder degree theory and a version of Leray-Schauder continuation theorem, we show that the system has a connected set of positive solutions for λ small.
1. Introduction In this paper, we study positive solutions of superlinear elliptic systems coupled by reaction terms involving product of powers. For motivation, we consider a system of the form α11 α12 in Ω; −∆w1 = w1 w2 α21 α22 (1.1) in Ω; −∆w2 = w1 w2 w1 = 0 = w2 on ∂Ω ,
where Ω is a bounded domain in RN (N ≥ 2) with C 2,η boundary ∂Ω, for some η ∈ (0, 1) (a bounded open
interval if N = 1). The exponents αij (i, j = 1, 2) satisfy the following assumptions: (A1) 0 ≤ α11 , α22 < 1;
(A2) α12 , α n21 >o0, α12 α21 > (1 − α11 )(1 − α22 ); and def def 2(b p+1) βb = (A3) max α b, βb > N − 1, where α b = (b p qb−1)+ , def
pb =
(N + 1)α12 N + 1 − (N − 1)α11
2(b q +1) (b p qb−1)+ def
and qb =
with
(N + 1)α21 . N + 1 − (N − 1)α22
It was shown in [16, Thm. 1.4 (i)] that under hypotheses (A1)-(A3), any eventual positive very weak solution (see [16] for definition) of the system (1.1) has uniform L∞ estimate and, moreover, the system (1.1) has a positive strong solution (w1 , w2 ) (to be defined shortly). Note that uniqueness of (w1 , w2 ) may not hold in general, even for scalar case, cf. [7]. Let (w1 , w2 ) be a positive strong solution of (1.1) and for λ > 0, define def
(z1 , z2 ) = (λ−θ1 w1 , λ−θ2 w2 ), where θ1 , θ2 > 0 are given by (1.2)
def
θ1 =
α12 + 1 − α22 α21 + 1 − α11 def > 0 and θ2 = > 0. α12 α21 − (1 − α11 ) (1 − α22 ) α12 α21 − (1 − α11 ) (1 − α22 )
2010 Mathematics Subject Classifications: 35J60, 35B32, 35B09, 35J47, 35J57. Key words: Elliptic systems; superlinear; positive solutions; continuum; bifurcation from infinity. The first author would like to acknowledge the project LO1506 of the Czech Ministry of Education, Youth and Sports for supporting her visit at the research centre NTIS - New Technologies for the Information Society of the Faculty of Applied Sciences, University of West Bohemia. The second author was supported by the project LO1506 of the Czech Ministry of Education.
1
2
M. CHHETRI AND P. GIRG
Then θ1 and θ2 satisfy (1.3)
1 + θ1 − θ1 α11 − θ2 α12
=
0, and
(1.4)
1 + θ2 − θ1 α21 − θ2 α22
=
0,
and (z1 , z2 ) satisfies α11 α12 in Ω; −∆z1 = λ z1 z2 α21 α22 −∆z2 = λ z1 z2 in Ω; z1 = 0 = z2 on ∂Ω .
(1.5)
Therefore, for each positive (component-wise) strong solution (w1 , w2 ) of (1.1), (z1 , z2 ) is a positive strong solution of (1.5) for any λ > 0. Clearly, {(λ, (z1 , z2 )) : λ > 0} is a smooth curve of positive strong
solutions emanating from infinity at λ = 0 with the property that kz1 k∞ , kz2 k∞ → +∞ as λ → 0+ and kz1 k∞ , kz2 k∞ → 0 as λ → +∞. Here and throughout the paper, k · k∞ denotes the L∞ -norm. Using this observation as our foundation, the purpose of this paper is to investigate the range of the bifurcation parameter λ > 0 for which the following perturbed system α11 α12 + f (x, u, v)] in Ω; −∆u = λ [u v α α 21 22 (1.6) −∆v = λ [u v + g (x, u, v)] in Ω; u = 0 = v on ∂Ω,
has a positive strong solution. The perturbations f, g : Ω × [0, +∞) × [0, +∞) → R are Carath´eodory
functions which are lower order terms relative to uα11 v α12 and uα21 v α22 (see hypothesis (B)). Using the rescaling technique discussed above, we will show that (1.6) can be transformed to a system which approaches to a system like (1.1) in the limit (in a sense to be clarified below). Then using the qualitative properties of positive strong solutions of the limiting system, we will establish that the re-scaled system and hence (1.6) has a positive strong solution for λ > 0 small. We wish to point out that our method is independent of the signs of the nonlinearities at the origin. By a strong solution of (1.6), we mean a triplet (λ, (u, v)), with u, v ∈ W 2,σ (Ω) ∩ W01,σ (Ω) for some σ > 1 and λ > 0 that satisfies equations of (1.6) a.e. in Ω (cf. [11, p. 219]). If u, v ≥ 0 (or u, v > 0) a.e. in Ω, then
(λ, (u, v)) is called a nonnegative (or positive) strong solution. 2
2
We take R × [L∞ (Ω)] as our underlying space, where [L∞ (Ω)] is a Banach space endowed with the norm def
k(u, v)k∞ = kuk∞ + kvk∞ . We denote the solution set of (1.6) by def
2
D = {(λ, (u, v)) ∈ (0, +∞) × [L∞ (Ω)] : (λ, (u, v)) is a strong solution of (1.6)} . We say that λ∞ ∈ R is a bifurcation point from infinity for (1.6) if there exists a sequence {(λn , (un , vn ))}∞ n=1 ∈ D such that λn → λ∞ and k(un , vn )k∞ → +∞ as n → +∞. By a continuum of strong solutions of (1.6) we
mean a subset C ⊂ D which is closed and connected. We say that a continuum C bifurcates from infinity at λ∞ if there exists a sequence {(λn , (un , vn ))}∞ n=1 ⊂ C such that λn → λ∞ and k(un , vn )k∞ → +∞ as
n → +∞.
Now we state our result. Theorem 1.1. Let (A1)-(A3) hold. Suppose (B) there exist K > 0, 0 ≤ βij < αij (i, j = 1, 2) and h ∈ Lξ (Ω), ξ > N such that |f (x, s, t)| ≤ Ksβ11 tβ12 + h(x)
and
|g(x, s, t)| ≤ Ksβ21 tβ22 + h(x) for s, t ≥ 0 and for a.e. x ∈ Ω .
ELLIPTIC SYSTEMS
3
Then there exists λ0 > 0 such that for any λ ∈ (0, λ0 ], there exists a positive strong solution (λ, (u, v)) of (1.6) with the property that kuk∞ , kvk∞ → +∞ as λ → 0+ . Moreover, there exists a continuum C of positive strong solutions of (1.6) bifurcating from infinity at λ∞ = 0 such that λ takes all values in (0, λ0 ] along C . Hypothesis (B) of Theorem 1.1 is satisfied by the following pairs of simple perturbations f and g: f (x, s, t) ≡ K1 and g(x, s, t) ≡ K2 for any K1 , K2 ∈ R. Readers are referred to [13, Thm. 1.6], [15, Thm. 2] and [16, Thm. 1.4(i)], where existence of a positive solution was discussed for (1.1). For excellent survey of results for general superlinear elliptic systems, for the case λ = 1, see [8, 9]. There are several existence results for (1.6) when λ > 0 and α11 = α22 = 0, see for example [3, 4, 5, 12, 17]. Theorem 1.1 complements these existence results by establishing the existence of a connected set of positive strong solutions of (1.6) bifurcating from infinity at λ∞ = 0. We prove Theorem 1.1 in Section 2. To prove our existence result, we use the Leray-Schauder degree theory combined with a re-scaling argument and degree computation for (1.5), see [16, Thm. 1.4(i)]. Similar ideas were employed in [1, 2, 10] for the scalar case and [3, 4] for the case of systems of equations. 2. Proof of Theorem 1.1 We extend the right hand sides of (1.6) to R as even functions and observe that any nonnegative strong solution of α12 α11 + f (x, |u|, |v|) in Ω; −∆u = λ |u| |v| (2.7) −∆v = λ |u|α21 |v|α22 + g (x, |u|, |v|) in Ω; u = 0 = v on ∂Ω, def
is a nonnegative strong solution of (1.6). For λ > 0, we re-scale (u, v) as (wλ,1 , wλ,2 ) = (λθ1 u, λθ2 v), where θ1 , θ2 > 0 satisfy (1.2). Using (1.3), we see that wλ,1 satisfies −∆wλ,1 = −∆(λθ1 u) = λθ1 (−∆u) = λ1+θ1 |u|α11 |v|α12 + f (x, |u|, |v|) = λ1+θ1 (λ−θ1 |wλ,1 |)α11 (λ−θ2 |wλ,2 |)α12 + f (x, λ−θ1 |wλ,1 |, λ−θ2 |wλ,2 |) = λ1+θ1 −θ1 α11 −θ2 α12 |wλ,1 |α11 |wλ,2 |α12 + λ1+θ1 f (x, λ−θ1 |wλ,1 |, λ−θ2 |wλ,2 |) = |wλ,1 |α11 |wλ,2 |α12 + λ1+θ1 f (x, λ−θ1 |wλ,1 |, λ−θ2 |wλ,2 |) .
Similarly, using (1.4), wλ,2 satisfies −∆wλ,2 = |wλ,1 |α21 |wλ,2 |α22 + λ1+θ2 g(x, λ−θ1 |wλ,1 |, λ−θ2 |wλ,2 |) . Let f˜, g˜ : (0, +∞) × Ω × R2 → R be defined by def f˜(λ, x, s, t) = λ1+θ1 f (x, λ−θ1 |s|, λ−θ2 |t|)
def
and g˜(λ, x, s, t) = λ1+θ2 g(x, λ−θ1 |s|, λ−θ2 |t|) .
We have β11 < α11 and β12 < α12 , thanks to (B), and therefore it follows from (1.3) that 1 + θ1 − θ1 β11 − θ2 β12 > 0. Using the estimate of (B), we find that |λ1+θ1 f (x, λ−θ1 |s|, λ−θ2 |t|)| ≤ λ1+θ1 K (λ−θ1 |s|)β11 (λ−θ2 |t|)β12 + h(x) = λ1+θ1 −θ1 β11 −θ2 β12 K |s|β11 |t|β12 + λ1+θ1 h(x) → 0
4
M. CHHETRI AND P. GIRG
and similarly, using (1.4), |λ1+θ2 g(x, λ−θ1 |s|, λ−θ2 |t|)| → 0 as λ → 0+ for any s, t ∈ R and pointwise a.e. in def def Ω. This allows us to extend f˜ and g˜ continuously to λ = 0 by setting f˜(0, x, s, t) = 0 and g˜(0, x, s, t) = 0. Therefore, for λ ≥ 0, the re-scaled function (wλ,1 , wλ,2 ) satisfies α12 α11 + f˜ (λ, x, wλ,1 , wλ,2 ) −∆wλ,1 = |wλ,1 | |wλ,2 | α α 22 21 (2.8) + g˜ (λ, x, wλ,1 , wλ,2 ) −∆wλ,2 = |wλ,1 | |wλ,2 | wλ,1 = 0 = wλ,2 on ∂Ω
in Ω; in Ω;
and for λ = 0, the system (2.8) reduces to the limiting system (by writing w0,1 = w1 and w0,2 = w2 ) α12 α11 in Ω; −∆w1 = |w1 | |w2 | α α 21 22 (2.9) −∆w2 = |w1 | |w2 | in Ω; w1 = w2 = 0 on ∂Ω .
Our aim is to show that the Leray-Schauder degree of the operator corresponding to the system (2.8) is nonzero on a suitable set, which in turn will establish the existence of a solution for (1.6). To do so, for 2
2
λ ≥ 0, we define the map Sλ : [L∞ (Ω)] → [L∞ (Ω)] by def Sλ (wλ,1 , wλ,2 ) = (−∆)−1 |wλ,1 |α11 |wλ,2 |α12 + f˜ (λ, x, wλ,1 , wλ,2 ) , |wλ,1 |α21 |wλ,2 |α22 + g˜ (λ, x, wλ,1 , wλ,2 ) ,
where (−∆)−1 : L∞ (Ω) ֒→ Lξ (Ω) → W 2,ξ (Ω) ∩ W01,ξ (Ω) ֒→֒→ L∞ (Ω) stands for the solution operator (in ˆ in Ω with z = 0 on ∂Ω and h ˆ ∈ L∞ (Ω), the strong sense) corresponding to the Dirichlet problem: −∆z = h see [11, Thm. 9.15, p. 241] . Then by the continuity of Nemytskii operators involved in the definition of Sλ
and the compactness of (−∆)−1 , the mapping (λ, (w1 , w2 )) 7→ Sλ (w1 , w2 ) is continuous and compact from [0, +∞) × [L∞ (Ω)]2 to [L∞ (Ω)]2 . Clearly solutions of (2.8) are solutions of the operator equation Sλ (wλ,1 , wλ,2 ) = (wλ,1 , wλ,2 ) or (I − Sλ )(wλ,1 , wλ,2 ) = (0, 0) ,
(2.10)
where I stands for the identity operator on [L∞ (Ω)]2 . Since I − Sλ is a compact perturbation of the identity,
the use of the Leray-Schauder degree is justified.
def def (u, v) ∈ [L∞ (Ω)]2 : k(u, v)k∞ < a and let 0 = (0, 0) ∈ [L∞ (Ω)]2 for For any a > 0, we set Ba = convenience. First, we compute the degree of I − Sλ corresponding to λ = 0, that is of I − S0 . Lemma 2.1. There exist R, r > 0 with R > r > 0 such that (2.11)
(w1 , w2 ) 6= S0 (w1 , w2 ) whenever k(w1 , w2 )k∞ = {r, R} , and
(2.12)
deg(I − S0 , BR \ Br , 0) = −1 .
Proof. For proof, see [16, proof of Thm. 1.4, pp. 77–78], where the fixed-point index on the cone of positive solutions was used. Since the right-hand sides of (2.9) are non-negative, we can repeat the same argument using Leray-Schauder degree. Now we use λ ≥ 0 as a homotopy parameter to determine the degree of I − Sλ . In particular, utilizing
the homotopy invariance of degree with respect to the homotopy parameter λ, the following lemma connects strong solutions of the limiting problem (2.9) to strong solutions of (2.8), and that solutions obtained are indeed positive. Lemma 2.2. Let R > r > 0 be as in Lemma 2.1. Then there exists λ0 > 0 such that (i) (wλ,1 , wλ,2 ) 6= Sλ (wλ,1 , wλ,2 ) for all λ ∈ [0, λ0 ] whenever k(wλ,1 , wλ,2 )k∞ = {r, R},
(ii) deg(I − Sλ , BR \ Br , 0) = −1 for all λ ∈ [0, λ0 ], and (iii) if (wλ,1 , wλ,2 ) = Sλ (wλ,1 , wλ,2 ) with λ ∈ [0, λ0 ] and r < k(wλ,1 , wλ,2 )k∞ < R, then (wλ,1 , wλ,2 ) is positive in Ω.
ELLIPTIC SYSTEMS
5
Proof. We prove part (i) by contradiction. Suppose there exists a sequence (λn , (wλn ,1 , wλn ,2 )) such that (wλn ,1 , wλn ,2 ) = Sλn (wλn ,1 , wλn ,2 ), k(wλn ,1 , wλn ,2 )k∞ ∈ {r, R} for each n ∈ N, and λn → 0+ as n → +∞. On
one hand, the compactness of (−∆)−1 : L∞ (Ω) → L∞ (Ω) implies that there exists (w1∗ , w2∗ ) ∈ [L∞ (Ω)]2 such that (wλn ,1 , wλn ,2 ) → (w1∗ , w2∗ ) in [L∞ (Ω)]2 (up to a subsequence) and therefore k(w1∗ , w2∗ )k∞ ∈ {r, R}. On
the other hand, Sλn (wλn ,1 , wλn ,2 ) → S0 (w1∗ , w2∗ ) (by continuity of Sλ ), a contradiction to (2.11). Therefore, there exists λ0 > 0 such that (i) holds. Using (i), the homotopy invariance of degree with respect to the homotopy parameter λ ∈ [0, λ0 ] and Lemma 2.1, the conclusion of (ii) follows, since deg(I − Sλ , BR \ Br , 0) = deg(I − S0 , BR \ Br , 0) = −1,
λ ∈ [0, λ0 ] .
Now we will prove part (iii). Suppose sα11 tα12 + f (x, s, t) ≥ 0 and sα21 tα22 + g(x, s, t) ≥ 0 for all s, t ≥ 0
and for a.e. x ∈ Ω, and strict inequalities hold on a subset of Ω of positive measure. Then since θ1 , θ2 are positive, it follows that the right hand side of (2.8) are positive as well and hence by maximum principle, (wλ,1 , wλ,2 ) is positive in Ω whenever r < k(wλ,1 , wλ,2 )k∞ < R with λ ∈ [0, λ0 ]. In the alternative case, we cannot apply maximum principle to the system (2.8). Instead, we will take advantage of the qualitative properties of solutions of the limiting problem (2.9). In particular, we observe that any strong solution of (2.9) is also a weak solution (in the usual sense). Using the strong comparison principle for weak solutions, see e.g. [6, Thm. 2.1] for p = 2, to each equation of (2.9) (combined with elliptic regularity), we see that any strong nontrivial nonnegative solution (w1 , w2 ) of (2.9) must be positive in Ω and ∂w2 ∂w1 < 0, < 0 on ∂Ω , ∂ν ∂ν where ν is the outward unit normal to ∂Ω. Now, assume to the contrary that there exists a sequence (2.13)
(λn , (wλn ,1 , wλn ,2 )) of solutions of the operator equation (2.10) and xn ∈ Ω such that λn → 0+ as n → +∞,
and wλn ,1 (xn ) ≤ 0 or wλn ,2 (xn ) ≤ 0 for each n ∈ N. Following the argument as in the proof of [3, Lemma 2.2, (iii)] and noting that the convergence of (wλn ,1 , wλn ,2 ) is in [C 1,η (Ω)]2 , we arrive at a contradiction to (2.13). This establishes (iii) by making λ0 smaller, if necessary. Now we conclude the proof of Theorem 1.1. For each λ ∈ [0, λ0 ], there exists a positive strong solution (wλ,1 , wλ,2 ) ∈ BR \ Br of (2.8) by Lemma 2.2. This means that (1.6) has a positive strong so-
lution (u, v) = (λ−θ1 wλ,1 , λ−θ2 wλ,2 ) for each λ ∈ (0, λ0 ], where θ1 , θ2 > 0 are given by (1.2). Since k(wλ,1 , wλ,2 )k∞ > r, we see that k(u, v)k∞ → +∞ as λ → 0+ . It follows, without loss of generality, that kvk∞ → +∞ as λ → 0+ . It remains to show that kuk∞ → +∞ as λ → 0+ as well. Indeed, assume to the contrary that there exists a sequence (λn , (un , vn )) of strong solutions of (1.6) such that λn → 0+ ,
kun k∞ is bounded and (λθn1 un , λθn2 vn ) = (wλn ,1 , wλn ,2 ) ∈ BR \ Br solves (2.8). Then, by the compactness of (−∆)−1 , (wλn ,1 , wλn ,2 ) → (w1 , w2 ), up to a subsequence, where (w1 , w2 ) ∈ BR \ Br solves (2.9). Since
kun k∞ is bounded, it follows that kwλn ,1 k∞ = λθn1 kun k∞ → 0 as λn → 0+ . This implies that w1 = 0 a.e. in Ω, a contradiction to the fact that any solution (w1 , w2 ) ∈ BR \ Br of (2.9) is component-wise positive in Ω. This completes the proof of the first part of Theorem 1.1.
Finally, to establish the second part of Theorem 1.1, we use the following version of the Leray-Schauder continuation theorem (see [10] and [14]). Proposition 2.3. Let X be a Banach space, Y ⊂ X a bounded open set and [a, b] ⊂ R. Suppose T : def
[a, b] × Y → X is completely continuous (i.e. continuous and compact). Define Σ = {(µ, x) ∈ [a, b] × Y :
x = T (µ, x)} and assume that the following conditions hold: (a): Σ ∩ ([a, b] × ∂Y ) = ∅, and
6
M. CHHETRI AND P. GIRG
(b): deg(I − T (a, ·), Y, 0) 6= 0, then Σ contains a continuum C along which µ takes all values in [a, b]. Using (i) and (ii) of Lemma 2.2, it follows from Proposition 2.3 that there exists a continuum C˜ of positive strong solutions of (2.8) along which λ takes all values in [0, λ0 ]. This in turn implies, using the relation (u, v) = (λ−θ1 wλ,1 , λ−θ2 wλ,2 ), that there exists a continuum C ⊂ D of positive strong solutions of (1.6) bifurcating from infinity at λ∞ = 0 such that λ takes all values in (0, λ0 ] along C . This completes the proof of Theorem 1.1.
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Department of Mathematics and Statistics, The University of North Carolina at Greensboro, Greensboro, NC 27402. USA, E-mail: [email protected] NTIS - New Technologies for the Information Society, University of West Bohemia, Univerzitn´ı 8, CZ-30614 Plzeˇ n, Czech Republic, E-mail: [email protected]
M. Chhetri, P. Girg
PII: DOI: Reference:
S0893-9659(17)30236-7 http://dx.doi.org/10.1016/j.aml.2017.07.005 AML 5299
To appear in:
Applied Mathematics Letters
Received date : 22 May 2017 Revised date : 17 July 2017 Accepted date : 18 July 2017 Please cite this article as: M. Chhetri, P. Girg, Superlinear elliptic systems with reaction terms involving product of powers, Appl. Math. Lett. (2017), http://dx.doi.org/10.1016/j.aml.2017.07.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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SUPERLINEAR ELLIPTIC SYSTEMS WITH REACTION TERMS INVOLVING PRODUCT OF POWERS M. CHHETRI AND P. GIRG
Abstract. We consider a system of the form −∆u = λg1 (x, u, v)
where λ > 0 is a parameter, Ω ⊂
−∆v = λg2 (x, u, v)
RN (N
u=0=v
in
Ω;
in
Ω;
on
∂Ω,
≥ 2) is a bounded domain with sufficiently smooth boundary
∂Ω (a bounded open interval if N = 1). Here gi (x, s, t) : Ω × [0, +∞) × [0, +∞) → R (i = 1, 2) are
Carath´ eodory functions that exhibit superlinear growth at infinity involving product of powers of u and v. Using re-scaling argument combined with Leray-Schauder degree theory and a version of Leray-Schauder continuation theorem, we show that the system has a connected set of positive solutions for λ small.
1. Introduction In this paper, we study positive solutions of superlinear elliptic systems coupled by reaction terms involving product of powers. For motivation, we consider a system of the form α11 α12 in Ω; −∆w1 = w1 w2 α21 α22 (1.1) in Ω; −∆w2 = w1 w2 w1 = 0 = w2 on ∂Ω ,
where Ω is a bounded domain in RN (N ≥ 2) with C 2,η boundary ∂Ω, for some η ∈ (0, 1) (a bounded open
interval if N = 1). The exponents αij (i, j = 1, 2) satisfy the following assumptions: (A1) 0 ≤ α11 , α22 < 1;
(A2) α12 , α n21 >o0, α12 α21 > (1 − α11 )(1 − α22 ); and def def 2(b p+1) βb = (A3) max α b, βb > N − 1, where α b = (b p qb−1)+ , def
pb =
(N + 1)α12 N + 1 − (N − 1)α11
2(b q +1) (b p qb−1)+ def
and qb =
with
(N + 1)α21 . N + 1 − (N − 1)α22
It was shown in [16, Thm. 1.4 (i)] that under hypotheses (A1)-(A3), any eventual positive very weak solution (see [16] for definition) of the system (1.1) has uniform L∞ estimate and, moreover, the system (1.1) has a positive strong solution (w1 , w2 ) (to be defined shortly). Note that uniqueness of (w1 , w2 ) may not hold in general, even for scalar case, cf. [7]. Let (w1 , w2 ) be a positive strong solution of (1.1) and for λ > 0, define def
(z1 , z2 ) = (λ−θ1 w1 , λ−θ2 w2 ), where θ1 , θ2 > 0 are given by (1.2)
def
θ1 =
α12 + 1 − α22 α21 + 1 − α11 def > 0 and θ2 = > 0. α12 α21 − (1 − α11 ) (1 − α22 ) α12 α21 − (1 − α11 ) (1 − α22 )
2010 Mathematics Subject Classifications: 35J60, 35B32, 35B09, 35J47, 35J57. Key words: Elliptic systems; superlinear; positive solutions; continuum; bifurcation from infinity. The first author would like to acknowledge the project LO1506 of the Czech Ministry of Education, Youth and Sports for supporting her visit at the research centre NTIS - New Technologies for the Information Society of the Faculty of Applied Sciences, University of West Bohemia. The second author was supported by the project LO1506 of the Czech Ministry of Education.
1
2
M. CHHETRI AND P. GIRG
Then θ1 and θ2 satisfy (1.3)
1 + θ1 − θ1 α11 − θ2 α12
=
0, and
(1.4)
1 + θ2 − θ1 α21 − θ2 α22
=
0,
and (z1 , z2 ) satisfies α11 α12 in Ω; −∆z1 = λ z1 z2 α21 α22 −∆z2 = λ z1 z2 in Ω; z1 = 0 = z2 on ∂Ω .
(1.5)
Therefore, for each positive (component-wise) strong solution (w1 , w2 ) of (1.1), (z1 , z2 ) is a positive strong solution of (1.5) for any λ > 0. Clearly, {(λ, (z1 , z2 )) : λ > 0} is a smooth curve of positive strong
solutions emanating from infinity at λ = 0 with the property that kz1 k∞ , kz2 k∞ → +∞ as λ → 0+ and kz1 k∞ , kz2 k∞ → 0 as λ → +∞. Here and throughout the paper, k · k∞ denotes the L∞ -norm. Using this observation as our foundation, the purpose of this paper is to investigate the range of the bifurcation parameter λ > 0 for which the following perturbed system α11 α12 + f (x, u, v)] in Ω; −∆u = λ [u v α α 21 22 (1.6) −∆v = λ [u v + g (x, u, v)] in Ω; u = 0 = v on ∂Ω,
has a positive strong solution. The perturbations f, g : Ω × [0, +∞) × [0, +∞) → R are Carath´eodory
functions which are lower order terms relative to uα11 v α12 and uα21 v α22 (see hypothesis (B)). Using the rescaling technique discussed above, we will show that (1.6) can be transformed to a system which approaches to a system like (1.1) in the limit (in a sense to be clarified below). Then using the qualitative properties of positive strong solutions of the limiting system, we will establish that the re-scaled system and hence (1.6) has a positive strong solution for λ > 0 small. We wish to point out that our method is independent of the signs of the nonlinearities at the origin. By a strong solution of (1.6), we mean a triplet (λ, (u, v)), with u, v ∈ W 2,σ (Ω) ∩ W01,σ (Ω) for some σ > 1 and λ > 0 that satisfies equations of (1.6) a.e. in Ω (cf. [11, p. 219]). If u, v ≥ 0 (or u, v > 0) a.e. in Ω, then
(λ, (u, v)) is called a nonnegative (or positive) strong solution. 2
2
We take R × [L∞ (Ω)] as our underlying space, where [L∞ (Ω)] is a Banach space endowed with the norm def
k(u, v)k∞ = kuk∞ + kvk∞ . We denote the solution set of (1.6) by def
2
D = {(λ, (u, v)) ∈ (0, +∞) × [L∞ (Ω)] : (λ, (u, v)) is a strong solution of (1.6)} . We say that λ∞ ∈ R is a bifurcation point from infinity for (1.6) if there exists a sequence {(λn , (un , vn ))}∞ n=1 ∈ D such that λn → λ∞ and k(un , vn )k∞ → +∞ as n → +∞. By a continuum of strong solutions of (1.6) we
mean a subset C ⊂ D which is closed and connected. We say that a continuum C bifurcates from infinity at λ∞ if there exists a sequence {(λn , (un , vn ))}∞ n=1 ⊂ C such that λn → λ∞ and k(un , vn )k∞ → +∞ as
n → +∞.
Now we state our result. Theorem 1.1. Let (A1)-(A3) hold. Suppose (B) there exist K > 0, 0 ≤ βij < αij (i, j = 1, 2) and h ∈ Lξ (Ω), ξ > N such that |f (x, s, t)| ≤ Ksβ11 tβ12 + h(x)
and
|g(x, s, t)| ≤ Ksβ21 tβ22 + h(x) for s, t ≥ 0 and for a.e. x ∈ Ω .
ELLIPTIC SYSTEMS
3
Then there exists λ0 > 0 such that for any λ ∈ (0, λ0 ], there exists a positive strong solution (λ, (u, v)) of (1.6) with the property that kuk∞ , kvk∞ → +∞ as λ → 0+ . Moreover, there exists a continuum C of positive strong solutions of (1.6) bifurcating from infinity at λ∞ = 0 such that λ takes all values in (0, λ0 ] along C . Hypothesis (B) of Theorem 1.1 is satisfied by the following pairs of simple perturbations f and g: f (x, s, t) ≡ K1 and g(x, s, t) ≡ K2 for any K1 , K2 ∈ R. Readers are referred to [13, Thm. 1.6], [15, Thm. 2] and [16, Thm. 1.4(i)], where existence of a positive solution was discussed for (1.1). For excellent survey of results for general superlinear elliptic systems, for the case λ = 1, see [8, 9]. There are several existence results for (1.6) when λ > 0 and α11 = α22 = 0, see for example [3, 4, 5, 12, 17]. Theorem 1.1 complements these existence results by establishing the existence of a connected set of positive strong solutions of (1.6) bifurcating from infinity at λ∞ = 0. We prove Theorem 1.1 in Section 2. To prove our existence result, we use the Leray-Schauder degree theory combined with a re-scaling argument and degree computation for (1.5), see [16, Thm. 1.4(i)]. Similar ideas were employed in [1, 2, 10] for the scalar case and [3, 4] for the case of systems of equations. 2. Proof of Theorem 1.1 We extend the right hand sides of (1.6) to R as even functions and observe that any nonnegative strong solution of α12 α11 + f (x, |u|, |v|) in Ω; −∆u = λ |u| |v| (2.7) −∆v = λ |u|α21 |v|α22 + g (x, |u|, |v|) in Ω; u = 0 = v on ∂Ω, def
is a nonnegative strong solution of (1.6). For λ > 0, we re-scale (u, v) as (wλ,1 , wλ,2 ) = (λθ1 u, λθ2 v), where θ1 , θ2 > 0 satisfy (1.2). Using (1.3), we see that wλ,1 satisfies −∆wλ,1 = −∆(λθ1 u) = λθ1 (−∆u) = λ1+θ1 |u|α11 |v|α12 + f (x, |u|, |v|) = λ1+θ1 (λ−θ1 |wλ,1 |)α11 (λ−θ2 |wλ,2 |)α12 + f (x, λ−θ1 |wλ,1 |, λ−θ2 |wλ,2 |) = λ1+θ1 −θ1 α11 −θ2 α12 |wλ,1 |α11 |wλ,2 |α12 + λ1+θ1 f (x, λ−θ1 |wλ,1 |, λ−θ2 |wλ,2 |) = |wλ,1 |α11 |wλ,2 |α12 + λ1+θ1 f (x, λ−θ1 |wλ,1 |, λ−θ2 |wλ,2 |) .
Similarly, using (1.4), wλ,2 satisfies −∆wλ,2 = |wλ,1 |α21 |wλ,2 |α22 + λ1+θ2 g(x, λ−θ1 |wλ,1 |, λ−θ2 |wλ,2 |) . Let f˜, g˜ : (0, +∞) × Ω × R2 → R be defined by def f˜(λ, x, s, t) = λ1+θ1 f (x, λ−θ1 |s|, λ−θ2 |t|)
def
and g˜(λ, x, s, t) = λ1+θ2 g(x, λ−θ1 |s|, λ−θ2 |t|) .
We have β11 < α11 and β12 < α12 , thanks to (B), and therefore it follows from (1.3) that 1 + θ1 − θ1 β11 − θ2 β12 > 0. Using the estimate of (B), we find that |λ1+θ1 f (x, λ−θ1 |s|, λ−θ2 |t|)| ≤ λ1+θ1 K (λ−θ1 |s|)β11 (λ−θ2 |t|)β12 + h(x) = λ1+θ1 −θ1 β11 −θ2 β12 K |s|β11 |t|β12 + λ1+θ1 h(x) → 0
4
M. CHHETRI AND P. GIRG
and similarly, using (1.4), |λ1+θ2 g(x, λ−θ1 |s|, λ−θ2 |t|)| → 0 as λ → 0+ for any s, t ∈ R and pointwise a.e. in def def Ω. This allows us to extend f˜ and g˜ continuously to λ = 0 by setting f˜(0, x, s, t) = 0 and g˜(0, x, s, t) = 0. Therefore, for λ ≥ 0, the re-scaled function (wλ,1 , wλ,2 ) satisfies α12 α11 + f˜ (λ, x, wλ,1 , wλ,2 ) −∆wλ,1 = |wλ,1 | |wλ,2 | α α 22 21 (2.8) + g˜ (λ, x, wλ,1 , wλ,2 ) −∆wλ,2 = |wλ,1 | |wλ,2 | wλ,1 = 0 = wλ,2 on ∂Ω
in Ω; in Ω;
and for λ = 0, the system (2.8) reduces to the limiting system (by writing w0,1 = w1 and w0,2 = w2 ) α12 α11 in Ω; −∆w1 = |w1 | |w2 | α α 21 22 (2.9) −∆w2 = |w1 | |w2 | in Ω; w1 = w2 = 0 on ∂Ω .
Our aim is to show that the Leray-Schauder degree of the operator corresponding to the system (2.8) is nonzero on a suitable set, which in turn will establish the existence of a solution for (1.6). To do so, for 2
2
λ ≥ 0, we define the map Sλ : [L∞ (Ω)] → [L∞ (Ω)] by def Sλ (wλ,1 , wλ,2 ) = (−∆)−1 |wλ,1 |α11 |wλ,2 |α12 + f˜ (λ, x, wλ,1 , wλ,2 ) , |wλ,1 |α21 |wλ,2 |α22 + g˜ (λ, x, wλ,1 , wλ,2 ) ,
where (−∆)−1 : L∞ (Ω) ֒→ Lξ (Ω) → W 2,ξ (Ω) ∩ W01,ξ (Ω) ֒→֒→ L∞ (Ω) stands for the solution operator (in ˆ in Ω with z = 0 on ∂Ω and h ˆ ∈ L∞ (Ω), the strong sense) corresponding to the Dirichlet problem: −∆z = h see [11, Thm. 9.15, p. 241] . Then by the continuity of Nemytskii operators involved in the definition of Sλ
and the compactness of (−∆)−1 , the mapping (λ, (w1 , w2 )) 7→ Sλ (w1 , w2 ) is continuous and compact from [0, +∞) × [L∞ (Ω)]2 to [L∞ (Ω)]2 . Clearly solutions of (2.8) are solutions of the operator equation Sλ (wλ,1 , wλ,2 ) = (wλ,1 , wλ,2 ) or (I − Sλ )(wλ,1 , wλ,2 ) = (0, 0) ,
(2.10)
where I stands for the identity operator on [L∞ (Ω)]2 . Since I − Sλ is a compact perturbation of the identity,
the use of the Leray-Schauder degree is justified.
def def (u, v) ∈ [L∞ (Ω)]2 : k(u, v)k∞ < a and let 0 = (0, 0) ∈ [L∞ (Ω)]2 for For any a > 0, we set Ba = convenience. First, we compute the degree of I − Sλ corresponding to λ = 0, that is of I − S0 . Lemma 2.1. There exist R, r > 0 with R > r > 0 such that (2.11)
(w1 , w2 ) 6= S0 (w1 , w2 ) whenever k(w1 , w2 )k∞ = {r, R} , and
(2.12)
deg(I − S0 , BR \ Br , 0) = −1 .
Proof. For proof, see [16, proof of Thm. 1.4, pp. 77–78], where the fixed-point index on the cone of positive solutions was used. Since the right-hand sides of (2.9) are non-negative, we can repeat the same argument using Leray-Schauder degree. Now we use λ ≥ 0 as a homotopy parameter to determine the degree of I − Sλ . In particular, utilizing
the homotopy invariance of degree with respect to the homotopy parameter λ, the following lemma connects strong solutions of the limiting problem (2.9) to strong solutions of (2.8), and that solutions obtained are indeed positive. Lemma 2.2. Let R > r > 0 be as in Lemma 2.1. Then there exists λ0 > 0 such that (i) (wλ,1 , wλ,2 ) 6= Sλ (wλ,1 , wλ,2 ) for all λ ∈ [0, λ0 ] whenever k(wλ,1 , wλ,2 )k∞ = {r, R},
(ii) deg(I − Sλ , BR \ Br , 0) = −1 for all λ ∈ [0, λ0 ], and (iii) if (wλ,1 , wλ,2 ) = Sλ (wλ,1 , wλ,2 ) with λ ∈ [0, λ0 ] and r < k(wλ,1 , wλ,2 )k∞ < R, then (wλ,1 , wλ,2 ) is positive in Ω.
ELLIPTIC SYSTEMS
5
Proof. We prove part (i) by contradiction. Suppose there exists a sequence (λn , (wλn ,1 , wλn ,2 )) such that (wλn ,1 , wλn ,2 ) = Sλn (wλn ,1 , wλn ,2 ), k(wλn ,1 , wλn ,2 )k∞ ∈ {r, R} for each n ∈ N, and λn → 0+ as n → +∞. On
one hand, the compactness of (−∆)−1 : L∞ (Ω) → L∞ (Ω) implies that there exists (w1∗ , w2∗ ) ∈ [L∞ (Ω)]2 such that (wλn ,1 , wλn ,2 ) → (w1∗ , w2∗ ) in [L∞ (Ω)]2 (up to a subsequence) and therefore k(w1∗ , w2∗ )k∞ ∈ {r, R}. On
the other hand, Sλn (wλn ,1 , wλn ,2 ) → S0 (w1∗ , w2∗ ) (by continuity of Sλ ), a contradiction to (2.11). Therefore, there exists λ0 > 0 such that (i) holds. Using (i), the homotopy invariance of degree with respect to the homotopy parameter λ ∈ [0, λ0 ] and Lemma 2.1, the conclusion of (ii) follows, since deg(I − Sλ , BR \ Br , 0) = deg(I − S0 , BR \ Br , 0) = −1,
λ ∈ [0, λ0 ] .
Now we will prove part (iii). Suppose sα11 tα12 + f (x, s, t) ≥ 0 and sα21 tα22 + g(x, s, t) ≥ 0 for all s, t ≥ 0
and for a.e. x ∈ Ω, and strict inequalities hold on a subset of Ω of positive measure. Then since θ1 , θ2 are positive, it follows that the right hand side of (2.8) are positive as well and hence by maximum principle, (wλ,1 , wλ,2 ) is positive in Ω whenever r < k(wλ,1 , wλ,2 )k∞ < R with λ ∈ [0, λ0 ]. In the alternative case, we cannot apply maximum principle to the system (2.8). Instead, we will take advantage of the qualitative properties of solutions of the limiting problem (2.9). In particular, we observe that any strong solution of (2.9) is also a weak solution (in the usual sense). Using the strong comparison principle for weak solutions, see e.g. [6, Thm. 2.1] for p = 2, to each equation of (2.9) (combined with elliptic regularity), we see that any strong nontrivial nonnegative solution (w1 , w2 ) of (2.9) must be positive in Ω and ∂w2 ∂w1 < 0, < 0 on ∂Ω , ∂ν ∂ν where ν is the outward unit normal to ∂Ω. Now, assume to the contrary that there exists a sequence (2.13)
(λn , (wλn ,1 , wλn ,2 )) of solutions of the operator equation (2.10) and xn ∈ Ω such that λn → 0+ as n → +∞,
and wλn ,1 (xn ) ≤ 0 or wλn ,2 (xn ) ≤ 0 for each n ∈ N. Following the argument as in the proof of [3, Lemma 2.2, (iii)] and noting that the convergence of (wλn ,1 , wλn ,2 ) is in [C 1,η (Ω)]2 , we arrive at a contradiction to (2.13). This establishes (iii) by making λ0 smaller, if necessary. Now we conclude the proof of Theorem 1.1. For each λ ∈ [0, λ0 ], there exists a positive strong solution (wλ,1 , wλ,2 ) ∈ BR \ Br of (2.8) by Lemma 2.2. This means that (1.6) has a positive strong so-
lution (u, v) = (λ−θ1 wλ,1 , λ−θ2 wλ,2 ) for each λ ∈ (0, λ0 ], where θ1 , θ2 > 0 are given by (1.2). Since k(wλ,1 , wλ,2 )k∞ > r, we see that k(u, v)k∞ → +∞ as λ → 0+ . It follows, without loss of generality, that kvk∞ → +∞ as λ → 0+ . It remains to show that kuk∞ → +∞ as λ → 0+ as well. Indeed, assume to the contrary that there exists a sequence (λn , (un , vn )) of strong solutions of (1.6) such that λn → 0+ ,
kun k∞ is bounded and (λθn1 un , λθn2 vn ) = (wλn ,1 , wλn ,2 ) ∈ BR \ Br solves (2.8). Then, by the compactness of (−∆)−1 , (wλn ,1 , wλn ,2 ) → (w1 , w2 ), up to a subsequence, where (w1 , w2 ) ∈ BR \ Br solves (2.9). Since
kun k∞ is bounded, it follows that kwλn ,1 k∞ = λθn1 kun k∞ → 0 as λn → 0+ . This implies that w1 = 0 a.e. in Ω, a contradiction to the fact that any solution (w1 , w2 ) ∈ BR \ Br of (2.9) is component-wise positive in Ω. This completes the proof of the first part of Theorem 1.1.
Finally, to establish the second part of Theorem 1.1, we use the following version of the Leray-Schauder continuation theorem (see [10] and [14]). Proposition 2.3. Let X be a Banach space, Y ⊂ X a bounded open set and [a, b] ⊂ R. Suppose T : def
[a, b] × Y → X is completely continuous (i.e. continuous and compact). Define Σ = {(µ, x) ∈ [a, b] × Y :
x = T (µ, x)} and assume that the following conditions hold: (a): Σ ∩ ([a, b] × ∂Y ) = ∅, and
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M. CHHETRI AND P. GIRG
(b): deg(I − T (a, ·), Y, 0) 6= 0, then Σ contains a continuum C along which µ takes all values in [a, b]. Using (i) and (ii) of Lemma 2.2, it follows from Proposition 2.3 that there exists a continuum C˜ of positive strong solutions of (2.8) along which λ takes all values in [0, λ0 ]. This in turn implies, using the relation (u, v) = (λ−θ1 wλ,1 , λ−θ2 wλ,2 ), that there exists a continuum C ⊂ D of positive strong solutions of (1.6) bifurcating from infinity at λ∞ = 0 such that λ takes all values in (0, λ0 ] along C . This completes the proof of Theorem 1.1.
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Department of Mathematics and Statistics, The University of North Carolina at Greensboro, Greensboro, NC 27402. USA, E-mail: [email protected] NTIS - New Technologies for the Information Society, University of West Bohemia, Univerzitn´ı 8, CZ-30614 Plzeˇ n, Czech Republic, E-mail: [email protected]