On a nonlinear parametric Robin problem with a locally defined reaction

Nonlinear Analysis 185 (2019) 374–387

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Nonlinear Analysis www.elsevier.com/locate/na

On a nonlinear parametric Robin problem with a locally defined reaction Leszek Gasiński a ,∗,1 , Nikolaos S. Papageorgiou b a b

Department of Mathematics, Pedagogical University of Cracow, Podchorazych 2, 30-084 Cracow, Poland Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece

article

info

Article history: Received 11 January 2019 Accepted 28 March 2019 Communicated by V.D. Radulescu MSC: 35J20 35J60 58E05

abstract We consider a nonlinear nonhomogeneous Robin problem with a parametric potential term and a Carathéodory reaction which is only locally defined. We show that for all values of the parameter in an upper half line, the equation has at least three nontrivial smooth solutions, two of constant sign and the third nodal. Under a symmetry condition on the reaction, we show the existence of a sequence of nodal solutions converging to zero in C 1 (Ω ). © 2019 Elsevier Ltd. All rights reserved.

Keywords: Reaction locally defined Nonlinear regularity Extremal constant sign solution Nodal solutions Critical groups

1. Introduction Let Ω ⊂ RN be a bounded domain with a C 2 -boundary ∂Ω . In this paper we study the following parametric Robin problem { p−2 −div a(Du(z)) + λ|u(z)| u(z) = f (z, u(z)) in Ω , (Pλ ) p−2 ∂u u = 0 on ∂Ω , ∂na + β(z)|u| with λ ∈ R and 1 < p < +∞. In this problem a : RN −→ RN is continuous and strictly monotone (thus maximal monotone too) and satisfies certain other regularity and growth conditions listed in hypotheses H(a) in Section 2. These hypotheses provide a general framework in which we can fit many differential operators of interest such as ∗

Corresponding author. E-mail addresses: [email protected] (L. Gasiński), [email protected] (N.S. Papageorgiou). 1 The research was supported by the National Science Center of Poland under Project No. 2015/19/B/ST1/01169, and the H2020-MSCA-RISE-2018 Research and Innovation Staff Exchange Scheme Fellowship within the Project No. 823731CONMECH. https://doi.org/10.1016/j.na.2019.03.019 0362-546X/© 2019 Elsevier Ltd. All rights reserved.

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the p-Laplacian and the (p, q)-Laplacian (that is the sum of a p-Laplacian and a q-Laplacian). There is also p−2 a parametric potential term λ|u| u with λ ∈ R being the parameter. On the right hand side, the reaction term f (z, x) is a Carath´eodory function (that is, for all x ∈ R, z ↦−→ f (z, x) is measurable and for a.a. z ∈ Ω , x ↦−→ f (z, x) is continuous). The distinguishing feature of our work here is that f (z, ·) is only locally defined, that is, the restrictions we impose on f (z, ·) concern only its behaviour near zero. In the boundary ∂u denotes the conormal derivative of u corresponding to the map a. We interpret this directional condition ∂n a derivative using the nonlinear Green’s identity (see Gasi´ nski–Papageorgiou [3, p. 210]) and if u ∈ C 1 (Ω ), then ∂u = (a(Du), n)RN , ∂na with n(·) being the outward unit normal on ∂Ω . The boundary coefficient β(z) is nonnegative. The case β ≡ 0 is also included and corresponds to the Neumann problem. Using variational tools based on the critical point theory, together with truncation, perturbation and comparison techniques and critical groups (Morse theory), we show that for all λ in an upper half line, problem (Pλ ) has at least three nontrivial smooth solutions all with sign information. More precisely, we have one positive solution, a second which is negative and third one which is nodal (that is, sign changing). Under a symmetry condition on f (z, ·) we produce a whole sequence {un }n⩾1 of nodal solutions such as un → 0 in C 1 (Ω ). The starting point of this paper is the work of Motreanu–Motreanu–Papageorgiou [11], where the authors deal with the Neumann problem (that is, β ≡ 0), the differential operator is less general (for example, the important case of (p, q)-Laplacian is not covered) and the reaction term f (z, ·) is globally defined and it is assumed to be (p − 1)-superlinear near ±∞. A similar problem driven by the Laplacian plus an indefinite and unbounded potential term and with a superlinear reaction term, was studied by Papageorgiou–R˘a dulescu–Repovˇs [14]. There the emphasis is on positive solutions and the authors prove a bifurcationtype result describing the set of positive solutions as the parameter λ varies. Finally we mention the work of Gasi´ nski–Klimczak–Papageorgiou [2], Gasi´ nski–Papageorgiou [4–6] on Dirichlet (p, 2)-equations and of Papageorgiou–Winkert [17], Gasi´ nski–Papageorgiou [7] on Robin problems. Works [2] and [17] consider reaction terms which are only locally defined. However, their conditions on f (z, ·) are more restrictive. They assume that f (z, ·) has zeros of constant sign and this way they impose a kind of oscillatory behaviour for f (z, ·) near zero. In contrast our hypotheses on f (z, ·) are minimal and general. We only require the presence of a “concave” term near zero. 2. Mathematical background — Hypotheses The main spaces in the study of problem (Pλ ) are the Sobolev space W 1,p (Ω ), the Banach space C 1 (Ω ) and the boundary Lebesgue spaces Lq (∂Ω ), 1 ⩽ q ⩽ +∞. By ∥ · ∥ we denote the norm of W 1,p (Ω ) defined by ( )1 ∥u∥ = ∥u∥pp + ∥Du∥pp p ∀u ∈ W 1,p (Ω ). The Banach space C 1 (Ω ) is ordered with positive (order) cone C+ = {u ∈ C 1 (Ω ) : u(z) ⩾ 0 for all z ∈ Ω }. This cone has a nonempty interior given by D+ = {u ∈ C+ : u(z) > 0 for all z ∈ Ω }. On ∂Ω we consider the (N − 1)-dimensional Hausdorff (surface) measure σ. Using this measure, we can define in the usual way the boundary Lebesgue spaces Lq (∂Ω ), 1 ⩽ q ⩽ +∞. From the theory of Sobolev

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spaces, we know that there exists a unique continuous linear map γ ˆ : W 1,p (Ω ) → Lp (∂Ω ), known as the “trace map” such that γ ˆ(u) = u|∂Ω ∀u ∈ W 1,p (Ω ) ∩ C(Ω ). This way we extend the notion of “boundary values” to all Sobolev functions. The trace map is not surjective and we have 1 ,p

im γ ˆ = W p′ (∂Ω )

and

ker γ ˆ = W01,p (Ω )

q (with p1 + p1′ = 1). The trace map γ ˆ is compact into Lq (∂Ω ) for all q ∈ [1, (NN−1)p −p ), p < N and into L (Ω ) for all q ⩾ 1 if N ⩽ p. In what follows for the sake of notational simplicity, we drop the use of the trace map γ ˆ. All restrictions of Sobolev functions on ∂Ω are understood in the sense of traces. For x ∈ R, we set x± = max{±x, 0}. Then given a measurable function u : Ω → R, we set u± (z) = u(z)± for all z ∈ Ω . If u ∈ W 1,p (Ω ), then u± ∈ W 1,p (Ω ). Given u, v ∈ W 1,p (Ω ) with u ⩽ v, we set

[u, v] = {y ∈ W 1,p (Ω ) : u(z) ⩽ y(z) ⩽ v(z) for a.a. z ∈ Ω }. Let X be a Banach space, φ ∈ C 1 (X) and c ∈ R. We define Kφ = {u ∈ X : φ′ (u) = 0} (the critical set of φ), φc = {u ∈ X : φ(u) ⩽ c} (the sublevel set of φ at c). Given a topological pair (A, B) such that B ⊆ A ⊆ X, by Hk (A, B), k ⩾ 0, we denote the kth relative singular homology group for the pair (A, B) with integer coefficients. If u ∈ Kφ is isolated and φ(u) = c, then the critical groups of φ at u are defined by Ck (φ, u) = Hk (φc ∩ U, φc ∩ U \ {u}) ∀k ⩾ 0, where U is a neighbourhood of u such that Kφ ∩ φc ∩ U = {u}. The excision property of singular homology, implies that this definition is independent of the isolating neighbourhood U . Let l ∈ C 1 (0, +∞) with l(t) > 0 for all t > 0. We assume that 0<ˆ c⩽

tl′ (t) ⩽ c0 l(t)

and c1 tp−1 ⩽ l(t) ⩽ c2 (ts−1 + tp−1 )

for all t > 0, some 1 ⩽ s < p, c1 , c2 > 0. The hypotheses on the map a are the following: H(a): a(y) = a0 (|y|)y for all y ∈ RN with a0 (t) > 0 for all t > 0 and (i) a0 ∈ C 1 (0, +∞), t ↦−→ a0 (t)t is strictly increasing on (0, +∞), a0 (t)t → 0+ as t → 0+ and lim

t→0+

a′0 (t)t > −1; a0 (t)

(ii) there exists c3 > 0 such that |∇a(y)| ⩽ c3

l(|y|) |y|

∀y ∈ RN \ {0};

(iii) we have (∇a(y)ξ, ξ)RN ⩾

l(|y|) 2 |ξ| |y|

∀y ∈ RN \ {0}, ξ ∈ RN ;

L. Gasiński and N.S. Papageorgiou / Nonlinear Analysis 185 (2019) 374–387

(iv) if G0 (t) =

∫t 0

377

a0 (s)s ds, then there exist q ∈ (1, p) and c∗ > 0 such that lim sup t→0+

qG0 (t) ⩽ c∗ tq

and there exists c4 > 0 such that c4 tp ⩽ a0 (t)t2 − qG0 (t) ∀t ⩾ 0. Remark 2.1. Hypotheses H(a)(i), (ii) and (iii) come from the nonlinear regularity theory of Lieberman [10] and from the nonlinear maximum principle of Pucci–Serrin [18, pp. 111, 120]. Hypothesis H(a)(iv) serves the needs of our problem. However, as the Examples below illustrate, this condition is mild and it is satisfied in all cases of interest. We emphasize that this differential operator u ↦−→ div a(Du) is not homogeneous and this is a source of difficulties in the analysis of problem (Pλ ). The primitive G0 is strictly convex and strictly increasing. We set G(y) = G0 (|y|) for all y ∈ RN . We have that G is convex and ∇G(y) = G′0 (|y|)

y = a0 (|y|)y = a(y) ∀y ∈ RN \ {0}. |y|

So, G is the primitive of a. From the convexity of G we have ∀y ∈ RN .

G(y) ⩽ (a(y), y)RN

(2.1)

From hypotheses H(a), we deduce the following properties for the map a (see Papageorgiou–R˘adulescu [12]). Lemma 2.2. If hypotheses H(a)(i), (ii) and (iii) hold, then (a) a is continuous and strictly monotone (thus maximal monotone too); (b) there exists c5 > 0 such that s−1

|a(y)| ⩽ c5 (|y| (c) we have

(a(y), y)RN ⩾

+ |y|

p−1

c1 p |y| p−1

)

∀y ∈ RN ; ∀y ∈ RN .

From this lemma and (2.1), we infer the following growth estimates for the primitive G. Corollary 2.3.

If hypotheses H(a)(i), (ii) and (iii) hold, then there exists c6 > 0 such that c1 p p |y| ⩽ G(y) ⩽ c6 (1 + |y| ) p(p − 1)

∀y ∈ RN .

Example 2.4. The following maps satisfy hypotheses H(a) (see [12]): p−2 (a) a(y) = |y| y, with 1 < p < +∞. This map corresponds to the p-Laplacian. p−2 q−2 (b) a(y) = |y| y + |y| y, with 1 < q < p < +∞. This map corresponds to the (p, q)-Laplacian (that is, the sum of a p-Laplacian and a q-Laplacian). Such differential operators arise in the mathematical models of many physical processes. We infer to works of Wilhelmsson [19] (plasma physics) and of Zhikov [20] (double phase problems in elasticity theory). 2 p−2 (c) a(y) = (1 + |y| ) 2 y, with 1 < p < +∞. This map corresponds to the extended capillary differential operator. p−2 1 (d) a(y) = |y| y(1 + 1+|y| p ), with 1 < p < +∞. The differential operator corresponding to this map, arises in certain models of plasticity theory (see Fuchs–Li [1]).

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By A : W 1,p (Ω ) ↦−→ W 1,p (Ω )∗ we denote the nonlinear map defined by ∫ ⟨A(u), h⟩ = (a(Du), Dh)RN dz ∀u, h ∈ W 1,p (Ω ). Ω

This map is continuous, monotone, hence maximal monotone too. For the boundary coefficient β, we assume the following: H(β) : β ∈ C 0,α (∂Ω ) with 0 < α < 1 and β(z) ⩾ 0 for all z ∈ ∂Ω . Remark 2.5. When β ≡ 0, we have the Neumann problem. For the reaction f the hypotheses are the following: H(f ) : f : Ω × R −→ R is a Carath´eodory function such that f (z, 0) = 0 for a.a. z ∈ Ω and (i) for every ϱ > 0, there exists aϱ ∈ L∞ (Ω ) such that |f (z, x)| ⩽ aϱ (z)

for a.a. z ∈ Ω , all |x| ⩽ ϱ;

(ii) there exist τ ∈ (1, q) (see hypothesis H(a)(iv)), c7 > 0 and δ > 0 such that τ

c7 |x| ⩽ f (z, x)x ⩽ τ F (z, x) where F (z, x) =

∫x 0

for a.a. z ∈ Ω , all |x| ⩽ δ0 ,

f (z, s) ds;

p−2 (iii) for every ϱ > 0, there exists ξˆϱ > 0 such that for a.a. z ∈ Ω , the function x ↦−→ f (z, x) + ξˆϱ |x| x is nondecreasing on [−ϱ, ϱ].

Remark 2.6. Evidently the hypotheses on f (z, ·) are minimal and concern only its behaviour near zero. We point out that no sign condition is assumed. We introduce the following sets: L+ =

{λ ∈ R : problem (Pλ ) admits a positive solution},

L− =

{λ ∈ R : problem (Pλ ) admits a negative solution},

Sλ+ Sλ−

—

the set of positive solutions of (Pλ ),

—

the set of negative solutions of (Pλ ).

3. Solutions of constant sign In this section we show the nonemptiness of L+ and L− and we produce extremal elements for Sλ+ and (that is, a smallest element for Sλ+ and a biggest element of Sλ− ).

Sλ−

Proposition 3.1. If hypotheses H(a), H(β) and H(f ) hold, then (a) L+ ̸= ∅, L− ̸= ∅ and Sλ+ ⊆ D+ , Sλ− ⊆ −D+ ; (b) if λ ∈ L+ (respectively λ ∈ L− ), η > λ and uλ ∈ Sλ+ (respectively vλ ∈ Sλ− ), then η ∈ L+ (respectively η ∈ L− ) and we can find uη ∈ Sη+ (respectively vη ∈ Sη− ) such that uη ⩽ uλ (respectively vλ ⩽ vη ).

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379

Proof . We consider the following auxiliary Robin problem { −div a(Du(z)) + u(z)p−1 = 1 in Ω , ∂u p−1 = 0 on ∂Ω , u > 0. ∂na + β(z)u Exploiting the surjectivity of maximal monotone coercive operators (see Gasi´ nski–Papageorgiou [3, p. 319]) p−2 and the strict monotonicity of the map u ↦−→ A(u) + |u| u we see that the above auxiliary Robin problem admits a unique positive solution u ∈ W 1,p (Ω ). In fact, the nonlinear regularity theory (see Lieberman [10]) and the nonlinear maximum principle (see Pucci-Serrin [18, pp. 111, 120]), imply that u ∈ D+ . We set Nf (u)(·) = f (·, u(·)) and define M0 = ∥Nf (u)∥∞

and m0 = min u > 0 Ω

(see hypothesis H(f )(i) and recall that u ∈ D+ ). We set λ = 1 +

M0 p−1 . m0

− div a(Du) + λu(z)p−1 > M0 ⩾ f (z, u(z))

We have

for a.a. z ∈ Ω .

We introduce the Carath´eodory function f+ (z, x) defined by { f (z, x+ ) if x ⩽ u, f+ (z, x) = f (z, u(z)) if u < x. ∫x We set F+ (z, x) = 0 f+ (z, s) ds and consider the C 1 -functional ψ+ : W 1,p (Ω ) −→ R defined by ∫ λ 1 F+ (z, x) dz ∀u ∈ W 1,p (Ω ), ψ+ (u) = γ0 (u) + ∥u∥pp − p p Ω where

∫

∫

γ0 (u) =

pG(Du) dz +

(3.1)

(3.2)

p

β(z)|u| dσ.

Ω

∂Ω

From (3.2), Corollary 2.3 and since λ > 0, we see that ψ+ is coercive. Also, from the Sobolev embedding theorem and the compactness of the trace map, we see that ψ+ is sequentially weakly lower semicontinuous. So, by the Weierstrass–Tonelli theorem, we can find u0 ∈ W 1,p (Ω ) such that ψ+ (u0 ) = inf{ψ+ (u) : u ∈ W 1,p (Ω )}.

(3.3)

Hypothesis H(a)(iv) implies that given c8 > c∗ , we can find δ1 ∈ (0, min{m0 , δ0 }) such that c8 q |y| q

G(y) ⩽

∀|y| ⩽ δ1 .

(3.4)

Let u ∈ D+ and choose t ∈ (0, 1) small such that 0 < tu(z) ⩽ δ1 Then we have ψ+ (tu) ⩽

∀z ∈ Ω

c8 tq tp ∥Du∥qq + q p

∫

and t|Du(z)| ⩽ δ1 p

β(z)|u| dσ + ∂Ω

∀z ∈ Ω .

tp λ c7 ∥u∥pp − tτ ∥u∥ττ p τ

(see (3.4), (3.5) and hypothesis H(f )(ii)). Since 1 < τ < q < p, choosing t ∈ (0, 1) even smaller if necessary, we have that ψ+ (tu) < 0, so ψ+ (u0 ) < 0 = ψ+ (0)

(3.5)

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(see (3.3)), and hence u0 ̸= 0. From (3.3), we have ′ ψ+ (u0 ) = 0,

so ∫ ∫ p−2 p−2 ⟨A(u0 ), h⟩ + β(z)|u0 | u0 h dσ + λ |u0 | u0 h dz Ω ∂Ω ∫ 1,p = f+ (z, u0 )h dz ∀h ∈ W (Ω ).

(3.6)

Ω 1,p In (3.6) first we choose h = −u− (Ω ). Then 0 ∈W

c1 − p p ∥Du− 0 ∥p + λ∥u0 ∥p ⩽ 0 p−1 (see Lemma 2.2(c) and hypothesis H(β)), so p c9 ∥u− 0∥ ⩽0

for some c9 > 0 (recall that λ > 0) and hence u0 ⩾ 0,

u0 ̸= 0.

Also in (3.6) we choose h = (u0 − u)+ ∈ W 1,p (Ω ). We have ∫ ∫ + ⟨A(u0 ), (u0 − u)+ ⟩ + β(z)up−1 (u − u) dσ + λ u0p−1 (u0 − u) dz 0 0 ∂Ω Ω ∫ = f (z, u)(u0 − u)+ dz Ω ∫ ∫ ⩽ ⟨A(u), (u0 − u)+ ⟩ + β(z)up−1 (u0 − u)+ dσ + λ up−1 (u0 − u)+ dz ∂Ω

Ω

(see (3.2), (3.1) and use the nonlinear Green’s identity), so u0 ⩽ u. Thus, we have proved that u0 ∈ [0, u],

u0 ̸= 0.

(3.7)

From (3.2), (3.6) and (3.7) it follows that { ˆ 0 (z)p−1 = f (z, u0 (z)) −div a(Du0 (z)) + λu p−1 ∂u0 = 0 on ∂Ω . ∂na + β(z)u0

for a.a. z ∈ Ω ,

From (3.8) and Proposition 2.10 of Papageorgiou–R˘adulescu [13], we have that u0 ∈ L∞ (Ω ). Then the nonlinear regularity of Lieberman [10] implies that u0 ∈ C+ \ {0}. Let ϱ = ∥u∥∞ and let ξˆϱ > 0 be as postulated by hypothesis H(f )(iii). We have div (Du0 (z)) ⩽ (ξˆϱ + λ)u0 (z)p−1

for a.a. z ∈ Ω ,

so u0 ∈ D+ (by the nonlinear Hopf’s theorem; see Pucci-Serrin [18]). We have proved that ˆ ∈ L+ ̸= ∅ λ

and Sλ+ ⊆ D+

∀λ ∈ L+ .

(3.8)

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381

Similarly using v = −u ∈ −D+ and reasoning as above, we show that L− ̸= ∅

and Sλ− ⊆ −D+

∀λ ∈ L− .

(b) Consider λ ∈ L+ , η > λ and uλ ∈ Sλ+ ⊆ D+ . For ϱ = ∥uλ ∥∞ , let ξˆϱ > 0 be as postulated by hypothesis H(f )(iii). Let ϑ+ eodory function defined by η (z, x) be the Carath´ { f (z, x+ ) + (ξˆϱ + ξ)(x+ )p−1 if x ⩽ uλ (z), (3.9) ϑ+ η (z, x) = p−1 ˆ f (z, uλ (z)) + (ξϱ + ξ)uλ (z) if uλ (z) < x, ∫x 1 1,p ˆ+ with ξ > −η. We set Θη+ (z, x) = 0 ϑ+ (Ω ) −→ R defined η (z, s) ds and consider the C -functional ψη : W by ∫ η+ξ 1 p + ˆ ψη (u) = γ0 (u) + ∥u∥p − Θη+ (z, u) dz ∀u ∈ W 1,p (Ω ). p p Ω Since ξ > −η, from (3.9) and Corollary 2.3, we see that ψˆ+ is coercive. Also it is sequentially weakly lower η

semicontinuous. So, as above (see the proof of (a)), via the Weierstrass-Tonelli theorem, we obtain minimizer uη ∈ W 1,p (Ω ) of the functional ψˆη+ such that uη ∈ [0, uλ ],

uη ̸= 0,

so uη ∈ Sη+ ⊆ D+ (see (3.9)) and thus η ∈ L+ and uη ⩽ uλ . Similarly, if λ ∈ L− , λ < η and vλ ∈ Sλ− , then we show that η ∈ L− and there exists vη ∈ Sη− ⊆ D+ such that vλ ⩽ vη . □ So, according to Proposition 3.1(b), the sets L+ and L− are upper half lines. Let λ+ and λ− ∗ = inf L+ ∗ = inf L− . − We can also guarantee that λ+ ∗ , λ∗ > −∞, if we impose the following sign condition on f (z, ·):

H0 : f (z, x)x ⩾ 0 for a.a. z ∈ Ω , all x ⩾ 0 and if x ̸= 0, then the inequality is strict for all z ∈ Ωx ⊆ Ω with |Ωx |N > 0 (here | · |N denotes the Lebesgue measure on RN ). Proposition 3.2. (a) If hypotheses H(a), H(f ), H0 hold and β ≡ 0 (Neumann problem), then L+ , L− ⊆ (0, +∞). − (b) If hypotheses H(a), H(β), H(f ) and H0 hold, then λ+ ∗ , λ∗ > −∞. Proof . (a) Let λ ∈ L+ ̸= ∅ (see Proposition 3.1). Then we can find u ∈ Sλ+ ⊆ D+ and we have ∫ ∫ p−1 ⟨A(u), h⟩ + λ u h dz = f (z, u)h dz ∀h ∈ W 1,p (Ω ) Ω

Ω

(recall that β = 0). Choosing h ≡ 1 in (3.10), we obtain ∫ λ up−1 dz ⩾ 0 Ω

(see hypothesis H0 ), so λ ⩾ 0 and thus λ+ ∗ ⩾ 0. + Suppose that λ∗ = 0 and 0 ∈ L+ . Then from (3.10) and hypothesis H0 we have ∫ 0= f (z, u) dz > 0, Ω

a contradiction. Therefore L+ ⊆ (0, +∞).

(3.10)

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Similarly we show that L− ⊆ (0, +∞). (b) Let λ ∈ L+ . Then we can find u ∈ Sλ+ ⊆ D+ and we have ∫ ∫ ∫ ⟨A(u), h⟩ + β(z)up−1 h dσ + λ up−1 h dz = f (z, u)h dz ∂Ω

Ω

∀h ∈ W 1,p (Ω ).

(3.11)

Ω

Choosing h ≡ 1 and using hypothesis H0 , we have ∫ ∫ β(z)up−1 dσ + λ up−1 dz ⩾ 0, ∂Ω

Ω

so λc10 ∥u∥p−1 ⩾ −∥u∥p−1 , for some c10 > 0, thus λ⩾−

1 c10

and so λ+ ∗ > −∞. Similarly we show that λ− ∗ > −∞. □ From Papageorgiou–R˘ a dulescu–Repovˇs [15] (see the proof of Proposition 7), we have that Sλ+ ⊆ D+ (with λ ∈ L+ ) is downward directed (this means that if u1 , u2 ∈ Sλ+ , then we can find u ∈ Sλ+ such that u ⩽ u1 and u ⩽ u2 ) and Sλ− ⊆ −D+ (with λ ∈ L− ) is upward directed (this means that if v1 , v2 ∈ Sλ− , then we can find v ∈ Sλ− such that v1 ⩽ v and v2 ⩽ v). Using these facts, we can show that if λ ∈ L+ ∩ L− , then problem (Pλ ) admits extremal constant sign solutions (that is, there exist a smallest positive and a biggest negative solution for problem (Pλ )). These solutions will be useful in producing a nodal solution (see Section 4). Proposition 3.3. If hypotheses H(a), H(β) and H(f ) hold, then for every λ ∈ L+ problem (Pλ ) has a smallest positive solution uλ ∈ D+ (that is, uλ ⩽ u for all u ∈ Sλ+ ) and for every λ ∈ L− problem (Pλ ) has a biggest negative solution v λ ∈ −D+ (that is, v ⩽ v λ for all v ∈ Sλ− ). Proof . Since Sλ+ is downward directed, invoking Lemma 3.10 of Hu–Papageorgiou [8, p. 178], we can find a decreasing sequence {un }n⩾1 ⊆ Sλ+ such that inf Sλ+ = inf un , n⩾1

0 ⩽ un ⩽ u1

∀n ⩾ 1

(3.12)

(the partial order in Sλ+ and the infimum are defined by the pointwise inequality between functions of Sλ+ ). We have ∫ ∫ ⟨A(un ), h⟩ + β(z)unp−1 h dσ + λ up−1 n h dz ∂Ω Ω ∫ = f (z, un )h dz ∀h ∈ W 1,p (Ω ), n ⩾ 1. (3.13) Ω

In (3.13) we choose h = un ∈ W 1,p (Ω ). Then using Lemma 2.2(c), (3.12) and hypothesis H(f )(i), we see that the sequence {un }n⩾1 ⊆ W 1,p (Ω ) is bounded. (3.14)

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From (3.13), (3.14) and Proposition 2.10 of Papageorgiou–R˘adulescu [13], we can find c11 > 0 such that un ∈ L∞ (Ω ),

∥un ∥∞ ⩽ c11

∀n ⩾ 1.

Applying the nonlinear regularity theory of Lieberman [10], we can find α ∈ (0, 1) and c12 > 0 such that un ∈ C 1,α (Ω ),

∥un ∥C 1,α (Ω) ⩽ c12

∀n ⩾ 1.

(3.15)

Recall that the embedding C 1,α (Ω ) ⊆ C 1 (Ω ) is compact. Hence from (3.15) and the monotonicity of the sequence {un }n⩾1 , we have (3.16) un −→ uλ in C 1 (Ω ), with uλ ⩾ 0 (see (3.12)). Suppose that uλ = 0. Then with δ0 > 0 as in hypothesis H(f )(ii), from (3.16) we see that we can find n0 ∈ N such that 0 < un (z) ⩽ δ0 ∀z ∈ Ω , n ⩾ n0 , so f (z, un (z)) ⩾ c7 un (z)τ −1

for a.a. z ∈ Ω , n ⩾ n0

(see hypothesis H(f )(ii)). We consider the Carath´eodory function k(z, x) defined by { c7 (x+ )τ −1 + ξ(x+ )p−1 if x ⩽ un (z), k(z, x) = c7 un (z)τ −1 + ξun (z)p−1 if un (z) < x,

(3.17)

(3.18)

∫x with ξ > −λ, n ⩾ n0 . We set K(z, x) = 0 k(z, s) ds and consider the C 1 -functional ψ˜λ : W 1,p (Ω ) −→ R defined by ∫ λ+ξ 1 p ˜ ∥u∥p − K(z, u) dz ∀u ∈ W 1,p (Ω ). ψλ(u) = γ0 (u) + p p Ω As before, via a minimization, we obtain u ˜λ ∈ W 1,p (Ω ) such that ψ˜λ (˜ uλ ) = inf{ψ˜λ (u) : u ∈ W 1,p (Ω )} < 0 = ψ˜λ (0), so u ˜λ ̸= 0. Also, we have ψ˜λ′ (˜ uλ ) = 0, so ∫ ∫ ⟨A(˜ uλ ), h⟩ + β(z)˜ up−1 h dσ + (λ + ξ) u ˜λp−1 h dz λ ∂Ω Ω ∫ 1,p = k(z, u ˜λ )h dz ∀h ∈ W (Ω ). Ω

1,p In (3.19) first we choose h = −˜ u− (Ω ). Then λ ∈W

c1 p p ∥D˜ u− u− λ ∥p + (λ + ξ)∥˜ λ ∥p ⩽ 0 p−1 (see Lemma 2.2(c), hypothesis H(β) and (3.18)), so u ˜λ ⩾ 0, (since ξ > −λ).

u ˜λ ̸= 0

(3.19)

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Also in (3.19) we choose h = (˜ uλ − un )+ ∈ W 1,p (Ω ). Then we have ∫ + ⟨A(˜ u), (˜ uλ − un ) ⟩ + β(z)˜ uλp−1 (˜ uλ − un )+ dσ ∂Ω ∫ + (λ + ξ) u ˜p−1 uλ − un )+ dz λ (˜ Ω ∫ = (c7 uτn−1 + ξup−1 uλ − un )+ dz n )(˜ ∫Ω ⩽ (f (z, un ) + ξup−1 uλ − un )+ dz n )(˜ Ω ∫ = ⟨A(un ), (˜ uλ − un )+ ⟩ + β(z)unp−1 (˜ uλ − un )+ dσ ∂Ω ∫ p−1 un (˜ uλ − un )+ dz + (λ + ξ) Ω

(see (3.18), (3.17), recall that n ⩾ n0 and un ∈ Sλ+ ), so u ˜λ ⩽ un

∀n ⩾ n0 ,

which contradicts (3.16) since we have assumed that uλ = 0. Therefore uλ ̸= 0 and since ∫ ∫ ∫ p−1 ⟨A(uλ ), h⟩ + β(z)up−1 u h dσ + λ h dz = f (z, uλ )h dz ∀h ∈ W 1,p (Ω ) λ λ ∂Ω

Ω

Ω

(see (3.12) and (3.16)), so uλ ∈ Sλ+ ⊆ D+

and uλ = inf Sλ+ .

Similarly, since Sλ− is upward directed, we can find an increasing sequence {vn }n⩾1 ⊆ Sλ− ⊆ −D+ such that sup Sλ− = sup vn . n⩾1

reasoning as above, we produce v λ ∈ W

1,p

(Ω ) such that

v λ ∈ Sλ− ⊆ −D+

and v λ = sup Sλ− . □

Remark 3.4. Note that in the above proof u ˜λ is the unique solution of { −div a(Du(z)) + λu(z)p−1 = c7 u(z)τ −1 in Ω , ∂u p−1 = 0 on ∂Ω , u > 0. ∂na + β(z)u Hence u ˜λ ∈ D+ . 4. Nodal solutions In this section, using the extremal constant sign solutions from Proposition 3.3, we will produce a nodal (sign-changing) solution. The idea is simple. We concentrate our attention on the order interval [v λ , uλ ]. Any nontrivial solution of (Pλ ) in that interval distinct from v λ and uλ , will necessarily be nodal (recall that v λ and uλ are extremal constant sign solutions). To this end we introduce the Carath´eodory function f0 (z, x) defined by ⎧ p−2 ⎨ f (z, v λ (z)) + ξ|v λ (z)| vλ (z) if x < v λ (z), p−2 f0 (z, x) = (4.1) f (z, x) + ξ|x| x if v λ (z) ⩽ x ⩽ uλ (z), ⎩ f (z, uλ (z)) + ξuλ (z)p−1 if uλ (z) < x, with ξ > −λ.

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Also, we consider the positive and negative truncations of f0 (z, ·), namely the Carath´eodory functions f0± (z, x) = f0 (z, ±x± ) ∀(z, x) ∈ Ω × R.

(4.2)

∫x ∫x We set F0 (z, x) = 0 f0 (z, s) ds and F0± (z, x) = 0 f0± (z, s) ds and consider the C 1 -functionals φλ , φ± λ : W 1,p (Ω ) −→ R defined by ∫ λ+ξ 1 p ∥u∥p − F0 (z, u) dz ∀u ∈ W 1,p (Ω ), φλ (u) = γ0 (u) + p p Ω ∫ λ+ξ 1 p γ (u) + ∥u∥ − F0± (z, u) dz ∀u ∈ W 1,p (Ω ). φ± (u) = 0 p λ p p Ω Proposition 4.1. If hypotheses H(a), H(β) and H(f ) hold and λ ∈ L+ ∩ L− , then problem (Pλ ) admits a nodal solution yλ ∈ [v λ , uλ ] ∩ C 1 (Ω ). Proof . Using (4.1) and (4.2) as before we can show that Kφλ ⊆ [v λ , uλ ] ∩ C 1 (Ω ),

Kφ+ ⊆ [0, uλ ] ∩ C+ , λ

Kφ− ⊆ [v λ , 0] ∩ (−C+ ). λ

The extremality of uλ ∈ D+ and of v λ ∈ −D+ , imply that Kφλ ⊆ [v λ , uλ ] ∩ C 1 (Ω ),

Kφ+ = {0, uλ }, λ

Kφ− = {0, v λ }.

(4.3)

λ

On account of (4.3), we may assume that Kφλ is finite.

(4.4)

Otherwise we already have an infinity of smooth nodal solutions and so we are done. From (4.1) and (4.2) it is clear that φ+ λ is coercive. Also, it is sequentially weakly lower semicontinuous. 1,p So, we can find u ˆλ ∈ W (Ω ) such that 1,p φ+ uλ ) = inf{φ+ (Ω )} < 0 = φ+ λ (ˆ λ (u) : u ∈ W λ (0)

(4.5)

(see hypothesis H(f )(ii)), so u ˆλ ̸= 0. Since u ˆλ ∈ Kφ+ \ {0} (from (4.5)), from (4.3) it follows that λ

u ˆλ = uλ ∈ D+ .

(4.6)

Note that φλ |C+ = φ+ λ |C+ (see (4.1) and (4.2)). Then from (4.5) and (4.6) it follows that uλ ∈ D+ is a local C 1 (Ω )-minimizer of φλ , so uλ ∈ D+ is a local W 1,p (Ω )-minimizer of φλ

(4.7)

(see Papageorgiou–R˘ adulescu [13, Proposition 2.12]). Similarly, using φ− λ , we show that v λ ∈ −D+ is a local W 1,p (Ω )-minimizer of φλ .

(4.8)

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We may assume that φλ (v λ ) ⩽ φλ (uλ ) (the reasoning is the same if the opposite inequality holds, using this time (4.8) instead of (4.7)). On account of (4.7), (4.4) and Theorem 5.7.6 of Papageorgiou–R˘a dulescu–Repovˇs [16, p. 367], we can find ϱ ∈ (0, 1) small such that φλ (v λ ) ⩽ φλ (uλ ) < inf{φλ (u) : ∥u − uλ ∥ = ϱ} = mλ ,

∥v λ − uλ ∥ > ϱ.

(4.9)

Since φλ is coercive (see (4.1)), we have that φλ satisfies the Cerami condition.

(4.10)

Then (4.9) and (4.10) permit the use of the mountain pass theorem (see, for example, Gasi´ nski–Papageorgiou 1,p [3, p. 648]). So, we can find yλ ∈ W (Ω ) such that yλ ∈ Kφλ ⊆ [v λ , uλ ] ∩ C 1 (Ω ),

mλ ⩽ φλ (yλ )

(4.11)

(see (4.3) and (4.9)), so yλ ̸∈ {v λ , uλ }.

(4.12)

Since yλ is a critical point of mountain pass type, using Theorem 6.5.8 of Papageorgiou–R˘adulescu– Repovˇs [16, p. 431], we have C1 (φλ , yλ ) ̸= 0. (4.13) On the other hand hypothesis H(f )(ii) and Proposition 3.7 of Papageorgiou–R˘adulescu [12] imply that Ck (φλ , 0) = 0 ∀k ⩾ 0.

(4.14)

From (4.13), (4.12) and (4.14) we conclude that yλ ̸∈ {0, v λ , uλ }, so yλ ∈ [v λ , uλ ] ∩ C 1 (Ω ) is a nodal solution of (Pλ ).

□

Summarizing our findings, we can state the following theorem. Theorem 4.2. If hypotheses H(a), H(β) and H(f ) hold, then for all λ ∈ R in an upper half line, problem (Pλ ) has at least three nontrivial solutions uλ ∈ D+ , vλ ∈ −D+ and yλ ∈ [vλ , uλ ] ∩ C 1 (Ω ) nodal. If we introduce a symmetry condition of f (z, ·), we can have a whole sequence of nodal solution converging to 0 in C 1 (Ω ). The new hypotheses on f are the following: H(f)’: f : Ω × R −→ R is a Carath´eodory function such that for a.a. z ∈ Ω , f (z, 0) = 0, for some η > 0, f (z, ·) is odd on [−η, η] and hypotheses H(f )′ (i), (ii), (iii) are the same as the corresponding hypotheses H(f )(i), (ii), (iii). Proposition 4.3. If hypotheses H(a), H(β) and H(f )′ hold, then for all λ ∈ R in an upper half line, problem (Pλ ) has a sequence {un }n⩾1 ⊆ C 1 (Ω ) of nodal solutions such that un → 0 in C 1 (Ω ). Proof . Let V ⊆ W 1,p (Ω ) be a finite dimensional subspace. Then all norms on V are equivalent. So, we can find ϱv ∈ (0, 1] such that u ∈ V, ∥u∥ ⩽ ϱv =⇒ |u(z)| ⩽ δ0 ∀z ∈ Ω . (4.15) Note that on account of hypothesis H(a)(iv) and Corollary 2.3, we have q

p

G(y) ⩽ c13 (|y| + |y| ) ∀y ∈ RN ,

(4.16)

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for some c13 > 0. Then, if u ∈ V , ∥u∥ ⩽ ϱv , from (4.15) and (4.16), we have φ(u) ⩽ c14 ∥y∥q − c15 ∥y∥τ , for some c14 , c15 > 0 (recall that ϱv ⩽ 1, q < p). Since τ < q, choosing ϱv ∈ (0, 1] even smaller, we have φ|V ∩∂Bϱv < 0. Then we can apply the symmetric mountain pass theorem of Kajikiya [9] and produce a sequence {un }n⩾1 ⊆ Kφλ ⊆ [v λ , uλ ] ∩ C 1 (Ω ) such that un −→ 0 in W 1,p (Ω ). As before (see the proof of Proposition 3.3), using the nonlinear regularity theory and the compactness of the embedding C 1,α (Ω ) ⊆ C 1 (Ω ), we show that un −→ 0 in C 1 (Ω ). □ References [1] M. Fuchs, G. Li, Variational inequalities for energy functionals with nonstandard growth conditions, Abstr. Appl. Anal. 3 (1–2) (1998) 41–64. [2] L. Gasi´ nski, L. Klimczak, N.S. Papageorgiou, Nonlinear Dirichlet problems with no growth restriction on the reaction, Z. Anal. Anwend. 36 (2) (2017) 209–238. [3] L. Gasi´ nski, N.S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton, FL, 2006. [4] L. Gasi´ nski, N.S. Papageorgiou, Nonlinear elliptic equations with a jumping reaction, J. Math. Anal. Appl. 443 (2) (2016) 1033–1070. [5] L. Gasi´ nski, N.S. Papageorgiou, Asymmetric (p, 2)-equations with double resonance, Calc. Var. Partial Differ. Equ. 56 (3) (2017) 88, 23. [6] L. Gasi´ nski, N.S. Papageorgiou, Multiplicity theorems for (p, 2)-equations, J. Nonlinear Convex Anal. 18 (7) (2017) 1297–1323. [7] L. Gasi´ nski, N.S. Papageorgiou, Positive solutions for the Robin p-Laplacian problem with competing nonlinearities, Adv. Calc. Var. 12 (1) (2019) 31–56. [8] S. Hu, N.S. Papageorgiou, HandBook of Multivalued Analysis. Vol. I, Theory, Kluwer Academic Publishers, Dordrecht, 1997. [9] R. Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal. 225 (2) (2005) 352–370. [10] G.M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations, Comm. Partial Differential Equations 16 (1991) 311–361. [11] D. Motreanu, V.V. Motreanu, N.S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear Neumann eigenvalue problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10 (3) (2011) 729–755. [12] N.S. Papageorgiou, V.D. R˘ adulescu, Coercive and noncoercive nonlinear Neumann problems with indefinite potential, Forum Math. 28 (3) (2016) 545–571. [13] N.S. Papageorgiou, V.D. R˘ adulescu, Nonlinear nonhomogeneous Robin problems with superlinear reaction term, Adv. Nonlinear Stud. 16 (2016) 737–764. [14] N.S. Papageorgiou, V.D. R˘ adulescu, Positive solutions for parametric semilinear Robin problems with indefinite and unbounded potential, Math. Scand. 121 (2) (2017) 263–292. [15] N.S. Papageorgiou, V.D. R˘ adulescu, D.D. Repovˇs, Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential, Discrete Contin. Dyn. Syst. 37 (5) (2017) 2589–2618. [16] N.S. Papageorgiou, V.D. R˘ adulescu, D.D. Repovˇs, Nonlinear Analysis - Theory and Methods, Springer, Switzerland, 2019. [17] N.S. Papageorgiou, P. Winkert, Nonlinear Robin problems with a reaction of arbitrary growth, Ann. Mat. Pura Appl. (4) 195 (4) (2016) 1207–1235. [18] P. Pucci, J. Serrin, The Maximum Principle, Birkh¨ auser Verlag, Basel, 2007. [19] H. Wilhelmsson, Explosive instabilities of reaction–diffusion equations, Phys. Rev. A (3) 36 (2) (1987) 965–966. [20] V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR-Izv. 29 (1987) 33–66.