Constant sign and nodal solutions for

Constant sign and nodal solutions for

nonhomogeneous Robin boundary value problems with asymmetric reactions

Antonio Iannizzotto

, Monica Marras

and Nikolaos S. Papageorgiou

1Department of Mathematics and Computer Science, University of Cagliari, Viale L. Merello 92, 09123 Cagliari, Italy

2Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece

Received 28 January 2018, appeared 25 August 2018 Communicated by Patrizia Pucci

Abstract. We study a nonlinear, nonhomogeneous elliptic equation with an asymmetric reaction term depending on a positive parameter, coupled with Robin boundary con- ditions. Under appropriate hypotheses on both the leading differential operator and the reaction, we prove that, if the parameter is small enough, the problem admits at least four nontrivial solutions: two of such solutions are positive, one is negative, and one is sign-changing. Our approach is variational, based on critical point theory, Morse theory, and truncation techniques.

Keywords: non-linear elliptic equations, asymmetric reactions, variational methods.

2010 Mathematics Subject Classification: 35J20, 35J60, 58E05.

1 Introduction

We study the following nonlinear, nonhomogeneous Robin problem:







−diva(∇u) +ξ(x)|u|^p−2u =λg(x,u) + f(x,u) in Ω

∂u

∂n_a +β(x)|u|^p−2u=0 on ∂Ω. (1.1)

HereΩ⊂_R^N(N>1) is a bounded domain with aC²-boundary∂Ω,p>1, anda:R^N →_R^N is a continuous, monotone mapping (hence maximal monotone too) which satisfies certain growth and regularity conditions (see hypotheses Ha below). These conditions are mild enough to include in our framework many non-linear operators of interest, such as the p- Laplacian, the (p,q)-Laplacian, and the generalized mean curvature operator. The potential functionξ ∈ L^∞(_Ω)is indefinite (i.e., sign-changing, see hypothesisHξ).

BCorresponding author. Email: antonio.iannizzotto@unica.it

On the right-hand side, λ > 0 is a parameter and g,f : Ω×_R → _R are Carathéodory functions. We assume that for a.a. x ∈ _Ω the mapping g(x,·) is strictly (p−1)-sublinear at ±_∞ (concave nonlinearity), while f(x,·) exhibits an asymmetric behavior, being (p−1)- superlinear at+_∞and asymptotically(p−1)-linear at−_∞(see hypothesesH_g,H_f below). So, in the positive semiaxis we have a competition phenomenon between a concave and a convex nonlinearity, while in the negative semiaxis and in the particular case of the p-Laplacian the equation may be resonant with respect to the first eigenvalue.

In the boundary condition,∂u/∂na denotes the generalized normal derivative correspond- ing to the mappinga, namely the extension of

u 7→ ha(∇u),ni, u∈C¹(_Ω)

toW^1,p(_Ω)(ndenotes the outward unit normal to∂Ω). The boundary coefficientβ∈C^0,α(_∂Ω) is non-negative, and the special caseβ=0 corresponds to the Neumann problem (see hypoth- esisHβ below).

In this paper, using variational methods based on the critical point theory, together with suitable truncation/perturbation techniques and Morse theory, we prove that, forλ>0 small enough, problem (1.1) has at least four nontrivial solutions: two positive, one negative, and one nodal (see Theorem 2.4 below and the ensuing discussion for a short account on our method).

Recently, elliptic boundary value problems with asymmetric reactions were studied in [15,28,29] (semilinear Dirichlet problems with zero potential), [18] (semilinear Neumann problem with indefinite potential), [3,14,23,24] (semilinear Robin problems with indefinite potential). For nonlinear elliptic equations we mention [13,16] (Dirichlet problems driven by thep-Laplacian), [20,26] (Dirichlet problems driven by the(p, 2)-Laplacian).

Compared with the existing literature, our result is novel in a twofold sense: unlike most of the aforementioned works, our result proves existence of four nontrivial solutions with precise sign information; and it holds for a very general problem, incorporating Robin and Neumann boundary conditions and several nonlinear leading differential operators as special cases (the only exception is represented by [16], which provides four solutions but only for Dirichlet conditions and the p-Laplace operator).

For the sake of completeness, we mention some more results on nonlinear Robin problems (with symmetric reactions) contained in [19,21,25].

The paper has the following structure: in Section2we introduce our hypotheses and main result, and we also establish some preliminary results and notations; in Section3we deal with constant sign solutions; and in Section4we investigate extremal constant sign solutions and nodal solutions.

2 Hypotheses and main result

We start this section by introducing and commenting the precise hypotheses on all features of problem (1.1). We begin with the mapping a:

H_a a : R^N →_R^N is defined by a(y) =a₀(|y|)yfor all y ∈ _R^N with a₀ : R+ → _R+, and we set for allt>⁰

H0(t) =

Z _t

0 a0(τ)τdτ and for ally∈ _R^N H(y) =H₀(|y|). Moreover:

(i) a₀ ∈ C¹(0,+_∞), a₀(t)>0 for all t >0, t 7→ a₀(t)t is strictly increasing on(0,+_∞), and

tlim→0⁺a0(t)t=0, lim

t→0⁺

a⁰₀(t)t a₀(t) >−1;

(ii) there existsθ ∈C¹(0,+_∞)s.t. for allt >0 c06 ^θ

0(t)t

θ(t) 6^c1, c2t^p−16^θ(_t)6^c3(_t^σ−1+_t^p−1) (_c0,c1,c2,c3 >_{0, 1}6^σ< _p)_, and for ally∈_R^N\ {₀}

|∇a(y)|6^c4

θ(|y|)

|y| (c₄ >0); (iii) for ally,z∈_R^N,y6=0

h∇a(y)z,zi> ^θ(|y|)|z|²

|y| ^; (iv) there existsr∈ (1,p]s.t.t 7→ H₀(t^1r)is convex,

lim sup

t→0⁺

rH₀(t)

t^r 6^c5 (c₅ >0), and for allt>⁰

pH0(t)−a0(t)t²>−c6 (c6 >0).

Hypotheses Ha (i)–(iii) are dictated by the nonlinear regularity theory of [12] and the nonlinear maximum principle of [27]. Hypothesis H_a (iv) serves the needs of our problem but is general enough to include several cases of interest (see Example2.2below). As a whole, Ha implies that H₀ is strictly convex and increasing onR+, and His convex with H(0) =0,

∇H(0) =0, and ∇H(y) = a(y)for ally ∈ _R^N\ {0}, i.e., His the primitive of a. This, along with convexity, clearly implies for all y∈_R^N

H(y)6ha(y),yi. (2.1)

HypothesesHa (i)–(iii)and (2.1) lead to the following properties ofaand H.

Lemma 2.1. IfH_a (i)–(iii)hold, then

(i) a:R^N →_R^N is continuous and monotone (hence maximal monotone);

(ii) |a(y)|6^c7(1+|y|^p−1)for all y∈ _R^N (c₇ >0);

(iii) ha(y),yi> ^c_p^2|₋^y|₁^{p for all y}∈_R^N;

(iv) _p^c₍²_p^|y₋^|₁^p₎ 6 ^H(y)6^c8(1+|y|^p)for all y∈_R^N (c₈ >0).

In what follows we shall denoteA :W^1,p(_Ω)→W^1,p(_Ω)^∗ the nonlinear differential oper- ator defined for allu,v ∈W^1,p(_Ω)_by

hA(u),vi=

Z

Ωha(∇u),∇vidx,

which is well defined by virtue ofH_a (ii). Such operator enjoys the(S)₊-property, i.e., when- ever(un)is a sequence inW^1,p(_Ω)s.t. un*uinW^1,p(_Ω)and

lim sup

n

hA(un),un−ui6^0,

thenu_n →uinW^1,p(_Ω)(see [17, p. 405]). Here follow some examples.

Example 2.2. The following mapsa:R^N →_R^N satisfyHa: (a) a(y) =|y|^p−2y, corresponding to the p-Laplace operator

∆pu=div(|∇u|^p−2∇u);

(b) a(y) =|y|^p−2y+|y|^q−2y(1< q< p< +_∞), corresponding to the(p,q)-Laplace operator

∆pu+_∆qu=div |∇u|^p−2∇u+|∇u|^q−2∇u .

Such operators arise in problems of mathematical physics, see [2] (reaction-diffusion equa- tions), [4] (elementary particles), [30] (plasma physics). Further:

(c) a(y) = (1+|y|²)^p−22y, corresponding to the generalized p-mean curvature operator div (1+|∇u|²)^p−22∇u

;

(d) a(y) = 2 ln(1+|y|^p) + (1+|y|^p)⁻¹y, corresponding to the operator div

2 ln(1+|∇u|^p)∇u+ ∇u 1+|∇u|^p

;

(e) a(y) =|y|^p−2y+|y|^p−2(1+|y|^p)⁻¹y, corresponding to the operator

∆pu+div

|∇u|^p−2∇u 1+|∇u|^p

.

Such operators arise in problems of nonlinear elasticity [7] and plasticity.

The other ingredients of (1.1) are subject to the following hypotheses:

Hξ ξ ∈ L^∞(_Ω);

H_β β∈C^0,α(_∂Ω)for someα∈ (0, 1),β(x)>^{0 for allx}∈ _∂Ω.

We note that the potential ξ may change sign, and that for β = 0 we recover the Neumann problem. Finally we introduce our hypotheses on the reactions, starting withg:

H_g g:Ω×_R→_Ris a Carathéodory function, for all(x,t)∈_Ω×_Rwe set G(x,t) =

Z _t

0 g(x,τ)dτ.

Moreover:

(i) for allρ >0 there existsa_ρ ∈ L^∞(_Ω)₊s.t. for a.a.x ∈_{Ω, all}|t|6^ρ

|g(x,t)|6^aρ(x); (ii) _{lim|t|→+∞} ^g(x,t)

|t|^p−2t =0 uniformly for a.a. x∈_Ω;

(iii) there existsq∈(1,r)s.t. for a.a. x∈ _{Ω, all}t∈_R

g(_x,t)_t >^c9|t|^q (_c9>₀)_; (iv) lim sup_t→0|^g_t⁽_|q^x,t−2⁾t 6^c10 uniformly for a.a. x∈_Ω(c₁₀ >0);

(v) there existsδ₀ >0 s.t. for a.a.x ∈_{Ω, all}|t|6^δ0

g(x,t)t6^qG(x,t).

We set ξ₀ = (p−1)ξ/c₂, β₀ = (p−1)β/c₂ (c₂ > 0 as in Ha (ii)) and we denote by ˆλ₁ = λˆ₁(p,ξ₀,β₀)>0 the first eigenvalue of the auxiliary problem







−_∆pu+ξ0(x)|u|^p−2u=λ|u|^p−2u in Ω

∂u

∂n_p +β₀(x)|u|^p−2u=0 on ∂Ω. (2.2) where

∂u

∂n_p =h|∇u|^p−2∇u,ni

(nbeing as usual the outward unit normal to∂Ω). Now we consider the asymmetric term f: H_f f :Ω×_R→_Ris a Carathéodory function, for all(x,t)∈_Ω×_Rwe set

F(x,t) =

Z _t

0 f(x,τ)dτ.

Moreover:

(i) for allρ >0 there existsb_ρ ∈ L^∞(_Ω)₊ s.t. for a.a.x ∈_{Ω, all}|t|6^ρ

|f(x,t)|6^bρ(x); (ii) lim_t→+_∞ ^f(x,t)

t^{p∗ −1} =0 uniformly for a.a. x∈ _Ω; (iii) limt→+_∞ ^f(x,t)

t^p−1 = +_∞uniformly for a.a.x∈ _Ω;

(iv) f(x,t)6^c11(t^p∗−1+t^r−1)−c₁₂t^p−1for a.a. x∈_{Ω, all}t >^{0 (c}11,c₁₂>_0);

(v) uniformly for a.a.x∈ _Ω

−c₁₃6^{lim inf}

t→−_∞

f(_x,t)

|t|^p−2t 6^{lim sup}

t→−_∞

f(_x,t)

|t|^p−2t 6 ^c2λˆ1

p−1 (c₁₃>0); (vi) _limt→0 ^f(x,t)

|t|^p−2t =0 uniformly for a.a. x∈_Ω;

(vii) there existsδ₁ >0 s.t. for a.a.x ∈_{Ω, all}|t|6^δ1

f(x,t)t >0.

Finally, we set for allλ>0 and all(x,t)∈_Ω×_R e_λ(x,t) =λg(x,t)t+ f(x,t)t−p

λG(x,t) +F(x,t) and we assume the following condition:

He for allλ>0

(i) there existsη_λ ∈ L¹(_Ω)₊s.t. for a.a. x∈ _{Ω, all 0}6^t6^t0 e_λ(x,t)6^eλ(x,t⁰) +η_λ(x); (ii) lim

t→−_∞e_λ(x,t) = +_∞uniformly for a.a.x∈_Ω.

We will writeHto mean all hypothesesHa,Hξ,Hβ,Hg,H_f, andHe.

By Hg (ii) g(x,·) is strictly (p−1)-sublinear at ±_∞, so it gives a ’concave’ contribution to the reaction of (1.1). Hypotheses H_f (iii), (v) imply that f(x,·) has an asymmetric be- havior at ±∞. More precisely, H_f (iii) means that f(x,·) is strictly (p−1)-superlinear at +_∞, so onR+ it represents a ’convex’ contribution to the reaction, leading to a competition phenomenon (concave-convex nonlinearities). We point out that the (p−1)-superlinearity of f(x,·) is not coupled with the usual Ambrosetti–Rabinowitz (AR) condition. Instead we use the less restrictive quasimonotonicity conditionHe (i), which includes in our framework (p−1)-superlinear reactions with a slower growth at+∞, that fail to satisfy(AR). Note that H_e (i)holds whenever we can find ρ>0 s.t. the mapping

t7→ ^λg(x,t) + f(x,t) t^p−1

is nondecreasing in [ρ,+_∞) for a.a. x ∈ _Ω [11]. On the negative semiaxis R−, by H_f (v) the mapping f(x,·) is asymptotically (p−1)-linear at −∞, and in the special case of the p-Laplacian (Example 2.2 (a) with c₂ = p−1) resonance with the principal eigenvalue is allowed. Resonance occurs from the left, so byHe(ii)problem (1.1) is coercive on the negative direction, which permits the use of the direct method of the calculus of variations. Finally we remark that byH_f (ii) f(x,·)does not satisfy the usual subcritical growth. Instead we have

’almost-critical’ growth, namely for allε >0 we can findc_ε >0 s.t. for a.a. x∈ _{Ω, all}t∈_R

|f(x,t)|6^ε|t|^p∗−1+cε.

This kind of growth is a source of technical difficulties, sinceW^1,p(_Ω)is not compactly em- bedded intoL^p∗(_Ω). We shall overcome such difficulties by using Vitali’s theorem.

Example 2.3. The following functions (of the typeλg+ f,λ > 0) satisfy hypothesesHg, H_f, andH_e:

(a) t7→ λ|t|^q−2t+

(λˆ₁|t|^p−2t ift 6⁰

t^s−1+t^r−1 ift >0 (q<r < p<s< p^∗); (b) t7→ λ|t|^q−2t+

(λˆ₁|t|^p−2t ift 60

t^p−1ln(1+t)−t^r−1 ift >0 (q<r < p);

(c) t 7→λ|t|^q−2t+







λˆ₁|t|^p−2t ift6⁰ t^p∗−1

ln(1+t^p)− ^pt

p^∗+p−1

p^∗(1+t^p)ln(1+t^p)² −t^r−1 ift >0 (q<r< p); Note that (a)satisfies (AR)while(b)does not, and that (c)has an almost critical growth at +_∞.

Our main result is the following.

Theorem 2.4. IfHhold, then there existsλ^∗ >0s.t. for allλ∈ (0,λ^∗)problem(1.1)admits at least four nontrivial solutions u+,v+,u−, ˜u∈ C¹(_Ω)with u+, v+positive in Ω, u−negative inΩ, andu˜ nodal.

Our approach is entirely variational, based on critical point theory. For allλ>0, we define an energy functional ϕ_λ for problem (1.1), which always admits 0 as a critical point. Then we introduce two truncated/perturbed functionals ˆϕ^±_λ, by replacing the indefinite potential ξ with a positive one and truncating the reactions at 0: thus, nontrivial critical points of ˆϕ⁺_λ (resp. ˆϕ⁻_λ) are positive (resp. negative) solutions of (1.1). Then we study separately the critical sets of such functionals: forλ>0 small enough, ˆϕ⁺_λ turns out to admit at least two nontrivial critical points, namely, a local minimizer v+ and another critical point u+ produced by the mountain pass theorem; while ˆϕ⁻_λ contributes a negative global minimizeru−.

Then we go further, proving existence of a smallest positive solution u₊ and a biggest negative solution u− of (1.1), and we truncate again the reactions. The resulting functional

˜

ϕ_λ selects solutions lying in the interval [u−, u₊], and admits a critical point ˜u of mountain pass type. By computing the critical groups at 0 and at ˜u, we see that ˜u6= 0, hence it must be nodal.

2.1 Notation

We establish some notation: we set R+ = [_0,+_∞)_{, R−} = (−_{∞, 0}]_; c₀,c₁, . . . denote positive constants; for allt ∈_Rwe set

t^±=max{0,±t}.

In any Banach space X, * denotes weak convergence and → strong convergence; if X is a function space on the domainD, then we denote the positive order cone by

X+={u∈ X: u(x)>^{0 for a.a. x}∈ D}.

We will say that a functional ϕ ∈ C¹(X)satisfies the Cerami condition (C), if any sequence (u_n)s.t. (ϕ(u_n))is bounded inR and (1+ku_nk)ϕ⁰(u_n) → 0 in X^∗, admits a (strongly) con- vergent subsequence. We will denote the set of critical points of ϕby

K(ϕ) ={u∈X: ϕ⁰(u) =₀}_.

We also recall the basic notion from Morse theory: let ϕ∈ C¹(X)andu∈K(ϕ)be an isolated critical point, namely there exists a neighborhood U ⊂ X of u s.t. K(ϕ)∩U = {u}, and ϕ(u) =c. Then, for allk∈_{N, the}k-th critical group of ϕatuis defined by

C_k(ϕ,u) =H_k {v∈U: ϕ(v)6^c},{v ∈U: ϕ(v)6^{c, v}6=u}, where H_k(·_,·)denotes thek-th singular homology group of a topological pair.

We shall use the function spaces(W^1,p(_Ω),k · k)and(C¹(_Ω),k · k_C1(_Ω)), endowed with the usual norms. Brackets h·,·i denote both the inner product of R^N and the duality between W^1,p(_Ω)^∗ andW^1,p(_Ω), with no possible confusing arising. We shall also use the Lebesgue spaces (L^ν(_Ω),k · k_ν) for all ν ∈ [1,+_∞], and the trace space (L^p(_∂Ω),k · k_Lp(_∂Ω)) (any u ∈ W^1,p(_Ω)will be identified with its trace on ∂Ω). We set

D+ ={u∈C¹(_Ω): u(x)>0 for allx ∈_Ω}, noting thatD+⊆int(C¹(_Ω)+).

3 Constant sign solutions

For allλ>0, u∈W^1,p(_Ω)we set ϕ_λ(u) =

Z

ΩH(∇u)dx+ ¹ p

Z

Ωξ(x)|u|^pdx+ ¹ p

Z

∂Ωβ(x)|u|^pdσ−

Z

Ω

λG(x,u) +F(x,u)dx

(the integral on∂Ωis computed with respect to the(N−1)-dimensional Hausdorff measure).

By Lemma2.1 (iv),H_ξ,H_β,Hg (i) (ii),H_f (i) (ii) (v), we have ϕ_λ ∈ C¹(W^1,p(_Ω)). Moreover, ϕ_λ is the energy functional for problem (1.1). Indeed, for allu∈K(ϕ_λ),v∈W^1,p(_Ω)we have

hA(u),vi+

Z

Ωξ(x)|u|^p−2uv dx+

Z

∂Ωβ(x)|u|^p−2uv dσ=

Z

Ω[λg(x,u) + f(x,u)]v dx, (3.1) i.e.,u is a (weak) solution of (1.1). Besides, let

µ>max

1, p−1 c₂

kξk_∞ (3.2)

and for all(x,t)∈_Ω×_Rset

k_λ(x,t) =λg(x,t) + f(x,t) +µ|t|^p−2t, k^±_λ(x,t) =kλ(x,±t^±),

and the primitives

K^(±)_λ (x,t) =

Z _t

0 k^(±)_λ (x,τ)dτ.

Now we define two truncated/perturbed functionals by setting for allu∈W^1,p(_Ω) ˆ

ϕ^±_λ(u) =

Z

ΩH(∇u)dx+ ¹ p

Z

Ω(ξ(x) +µ)|u|^pdx+ ¹ p

Z

∂Ωβ(x)|u|^pdσ−

Z

ΩK^±_λ(x,u)dx (note thatξ+µis positive by (3.2)). We shall study separately the properties of ˆϕ⁺_λ and ˆϕ⁻_λ, which are different by the asymmetry of f.

Lemma 3.1. IfHhold, then for allλ>0 ˆϕ⁺_λ ∈C¹(W^1,p(_Ω))satisfies(C).

Proof. Clearlyk⁺_λ :Ω×_R →_Ris a Carathéodory function, with a growth defined byH_g (ii), H_f (ii) (v)_{, so ˆ}ϕ⁺_λ ∈C¹(W^1,p(_Ω))_.

Let (u_n) be a sequence in W^1,p(_Ω) s.t. |ϕˆ⁺_λ(u_n)| 6 ^c14 for all n ∈ _N (c₁₄ > 0) and (1+ kunk)(ϕˆ⁺_λ)⁰(un) → 0 inW^1,p(_Ω)^∗. We can find a sequence (ε_n)in R s.t. ε_n → 0⁺ and for all n∈_N,v∈W^1,p(_Ω)

hA(u_n),vi+

Z

Ω(ξ(x) +µ)|u_n|^p−2u_nv dx+

Z

∂Ωβ(x)|u_n|^p−2u_nv dσ−

Z

Ωk⁺_λ(x,u_n)v dx 6 ^εnkvk

1+kunk^{. (3.3)}

Choosing v=−u⁻_n in (3.3) and using Lemma2.1(iii)we get for alln∈ _N c2

p−1k∇u⁻_nk^p_p+

Z

Ω(ξ(x) +µ)(u⁻_n)^pdx6^εn.

Passing to the limit we see that u⁻_n → 0 inW^1,p(_Ω). Now we deal with u⁺_n. By definition of K⁺_λ we have for all n∈_N

pc₁₄> ^p

Z

ΩH(∇un)dx+

Z

Ω(ξ(x) +µ)|un|^pdx+

Z

∂Ωβ(x)|un|^pdσ−p Z

ΩK⁺_λ(x,un)dx

> ^p

Z

ΩH(∇u⁺_n)dx+

Z

Ωξ(x)(u⁺_n)^pdx+

Z

∂Ωβ(x)(u⁺_n)^pdσ−p Z

Ω

λG(x,u⁺_n) +F(x,u⁺_n)dx, while (3.3) with v=−u⁺_n yields

−hA(u⁺_n),u⁺_ni −

Z

Ωξ(x)(u⁺_n)^pdx−

Z

∂Ωβ(x)(u⁺_n)^pdσ+

Z

Ω[λg(x,u⁺_n) + f(x,u⁺_n)]u⁺_n dx6^εn. Adding up we get

Z

Ω

pH(∇u⁺_n)− ha(∇u⁺_n),∇u⁺_nidx+

Z

Ωe_λ(x,u⁺_n)dx6^c15 (c₁₅>0), which byHa (iv)implies

Z

Ωe_λ(x,u⁺_n)dx6^c16 (c₁₆>0). (3.4) We claim that (u⁺_n)is bounded in W^1,p(_Ω). Arguing by contradiction, we may assume that (passing if necessary to a subsequence) ku⁺_nk → +∞. Then we set for all n ∈ _N wn = u⁺_nku⁺_nk⁻¹, so w_n∈W^1,p(_Ω)withkw_nk=1. Passing again to a subsequence we havew_n *w in W^1,p(_Ω) _and w_n → w both in L^p(_Ω) _{and in} L^p(∂Ω) (due to the compact embeddings W^1,p(_Ω),→ L^p(_Ω), L^p(_∂Ω)). Clearlyw∈W^1,p(_Ω)+. Two cases may occur:

(a) First we assumew6=0. Let

Ω+= {x∈_Ω: w(x)>0},

then |_Ω+|> 0 and for a.a. x ∈ _{Ω+ we have}u⁺_n(x)→ +_{∞. ByHf} (iii)we have for a.a.

x∈_Ω+

F(x,u⁺_n(x))

ku⁺_nk^p = ^F(x,u⁺_n(x))

u⁺_n(x)^{p wn}(x)^p →+_∞ By Fatou’s lemma we have

limn

Z

Ω⁺

F(x,u⁺_n)

ku⁺nk^{p dx}= +_∞.

ByH_f (i) (iii)we have for a.a.x∈ _{Ω, all}t>⁰

F(x,t)>^tp−c₁₇ (c₁₇>₀)_, so we have

Z

Ω\_Ω+

F(x,u⁺_n) ku⁺_nk^{p dx}>

Z

Ω\_Ω+

w_n^pdx− ^c17|_Ω| ku⁺_nk^p, and the latter is bounded from below. Summarizing,

limn

Z

Ω

F(x,u⁺_n)

ku⁺_nk^{p dx}= +_∞. (3.5)

Besides,Hg (ii)implies, as above, limn

Z

Ω⁺

G(x,u⁺_n)

ku⁺_nk^{p dx}=0.

ByHg (i) (ii), for anyε>0 we can findc₁₈=c₁₈(ε)>0 s.t. for a.a. x∈_{Ω, all}t>⁰ G(x,t)6 ^ε

pt^p+c₁₈. So we have

lim sup

n

Z

Ω\_Ω+

G(x,u⁺_n)

ku⁺_nk^{p dx}6^{lim sup}

n

Z

Ω\_Ω+

ε

pw_n^p+ ^c18 ku⁺_nk^p

dx6 ^ε pkwk^p_p. sinceε>0 is arbitrary, adding the two integrals we get

limn

Z

Ω

G(x,u⁺_n)

ku⁺_nk^{p dx} =0. (3.6)

Now (3.5), (3.6) imply limn

Z

Ω

λG(x,u⁺_n) +F(x,u⁺_n)

ku⁺_nk^{p dx} = +_∞.

But again from boundedness of (ϕˆ⁺_λ(u_n)), and recalling that u⁻_n → 0 in W^1,p(_Ω), we have for alln∈_N

Z

ΩH(∇u⁺_n)dx+ ¹ p Z

Ωξ(x)(u⁺_n)^pdx+¹ p Z

∂Ωβ(x)(u⁺_n)^pdσ−

Z

Ω

λG(x,u⁺_n) +F(x,u⁺_n)dx

>−c₁₉ (c₁₉>0),

which, along with Lemma2.1(iv), implies Z

Ω

λG(x,u⁺_n) +F(x,u⁺_n) ku⁺_nk^{p dx} 6 ^c19

ku⁺_nk^p +

Z

Ω

c₈(1+|∇u⁺_n|^p)

ku⁺_nk^{p dx}+ ¹ p

Z

Ωξ(x)w_n^pdx+ ¹ p

Z

∂Ωβ(x)w_n^pdσ 6^c20(1+kw_nk^p) (c₂₀>0),

and the latter is bounded from above. Thus we reach a contradiction.

(b) Now we assume w = 0. Fix M > 0 and set ˆw_n = (Mp)^1pw_n for all n ∈ _{N, so ˆ}w_n * 0 inW^1,p(_Ω)_{and ˆ}w_n →_{0 in} L^p(_Ω)_and L^p(∂Ω)_{. ByHg} (i) (ii)we have for a.a.x∈ _{Ω, all} t ∈_R

|G(x,t)|6^c21(1+|t|^p) (c₂₁ >0). So we get

limn

Z

ΩG(_{x, ˆwn})_dx=_{0. (3.7)}

Clearly(wˆ_n)is bounded inL^p∗(_Ω), so set K₀ =sup

n∈_N

kwˆnk^p_p^∗∗.

ByH_f (i) (ii), for anyε>0 we can findc₂₁=c₂₂(ε)>0 s.t. for a.a. x∈_{Ω, all}t∈_R

|F(x,t)|6 ^ε

2K₀|t|^p∗+c22.

So, the sequence (F(·_{, ˆ}w_n)) is bounded in L¹(_Ω). Furthermore, for any measurable set B⊂_Ωwith |B|6^ε(2c₂₂)⁻¹ we have for alln∈_N

Z

B

|F(x, ˆw_n)|dx6 ^ε

2K₀kwˆ_nk^p_p^∗∗+c₂₂|B|6^ε.

So the sequence(F(·, ˆw_n))is uniformly integrable inΩ(see [8, Problem 1.6]). Passing to a subsequence, we have F(x, ˆwn(x)) →0 asn → ∞, for a.a. x ∈ Ω. By Vitali’s theorem [8, p. 5] we have

limn

Z

ΩF(x, ˆw_n)dx=_{0. (3.8)}

Sinceku⁺_nk →+_{∞, for}n∈_Nbig enough we have 0< (Mp)^1p

ku⁺_nk 6^{1. (3.9)}

Let ˆψ⁺_λ ∈C¹(W^1,p(_Ω))be defined for allu ∈W^1,p(_Ω)by ψˆ⁺_λ(u) = ^c2

p(p−1)k∇uk^p_p+ ¹ p

Z

Ω(ξ(x) +µ)|u|^pdx+ ¹ p

Z

∂Ωβ(x)|u|^pdσ−

Z

ΩK⁺_λ(x,u)dx.

For alln∈_Nthere existstn∈[0, 1]s.t.

ψˆ_λ⁺(t_nu⁺_n) = max

t∈[0,1]

ψˆ⁺_λ(tu⁺_n). In particular, by (3.9) we have forn∈_Nbig enough

ψˆ⁺_λ(t_nu⁺_n)>^ψˆ+_λ(wˆ_n)

> ^c2M

p−1k∇wnk^p_p+M Z

Ω(ξ(x) +µ)w^p_ndx−

Z

Ω

λG(x, ˆwn) +F(x, ˆwn) +Mµw_n^p dx

>^M(c₂₃−µkw_nk^p_p)−

Z

Ω

λG(x, ˆw_n) +F(x, ˆw_n)dx (c₂₃>0)

(recall thatµ> kξk_∞ andkw_nk=1). Now by (3.7), (3.8) we have forn∈_Neven bigger ψˆ⁺_λ(t_nu⁺_n)> ^Mc24 (c₂₄ >₀)

which by arbitrarity of M > 0 implies ˆψ⁺_λ(t_nu⁺_n)→ +_∞as n → ∞. By Lemma 2.1 (iv) we have ˆϕ⁺_λ(u)>^ψˆ+_λ(u)for allu ∈W^1,p(_Ω), hence the sequence (ψ^ˆ_λ⁺(u⁺_n))is bounded from above. Besides, clearly ˆψ_λ⁺(0) = 0. So, for all n ∈ _N big enough we must have t_n∈(0, 1). By definition oft_n, then,

d

dt ψˆ_λ⁺(tu⁺_n)

t=tn =h(ψ^ˆ+_λ)⁰(t_nu⁺_n),u⁺_ni=0.

Multiplying byt_nwe get c₂

p−1k∇(t_nu⁺_n)k^p_p+

Z

Ωξ(x)(t_nu⁺_n)^pdx+

Z

∂Ωβ(x)(t_nu⁺_n)^pdσ

=

Z

Ω[λg(x,t_nu⁺_n) + f(x,t_nu⁺_n)]t_nu⁺_n dx.

ByHe (i), t_n <1, and (3.4) we have Z

Ωe_λ(x,t_nu⁺_n)dx6

Z

Ωe_λ(x,u⁺_n)dx+kη_λk₁< c₂₅ (c₂₅ >₀)_, which implies

Z

Ω[λg(x,t_nu⁺_n) + f(x,t_nu⁺_n)]t_nu⁺_n dx6 ^p

Z

Ω

λG(x,t_nu⁺_n) +F(x,t_nu⁺_n)dx+c₂₅. Thus, for alln∈_Nbig enough we have

pψˆ⁺_λ(t_nu⁺_n) =

Z

Ω[λg(x,t_nu⁺_n) + f(x,t_nu⁺_n)]t_nu⁺_n dx−p Z

Ω

λG(x,t_nu⁺_n) +F(x,t_nu⁺_n)dx 6^c25,

a contradiction.

By the claim above and u⁻_n → 0, we see that (un) is bounded in W^1,p(_Ω). Passing to a subsequence, we may assumeu_n * u inW^1,p(_Ω) andu_n → uin L^p(_Ω)and L^p(_∂Ω). ByH_g (i) (ii)we have for a.a.x ∈_Ω, allt ∈_R

|g(x,t)|6^c26(1+|t|^p−1) (c₂₆ >0), hence by Hölder’s inequality

limn

Z

Ωg(x,u⁺_n)(u_n−u)dx=0. (3.10) Besides,(u_n)is bounded inL^p∗(_Ω), so we set

K₁ =sup

n∈_N

ku_nk_p^∗+kuk_p^∗.

ByH_f (i) (ii)_{, for any}ε>0 we can findc₂₇= c₂₇(ε)>0 s.t. for a.a.x∈ _{Ω, all}t>0

|f(x,t)|6 ^ε

3K₁^p∗t^p∗−1+c₂₇.

Passing if necessary to a subsequence, we have f(x,u⁺_n(x))(u_n(x)−u(x))→0 as n→ _{∞, for} a.a.x ∈Ω. Moreover, for any measurableB⊂ _Ωwith

|B|6 ε

6K₁c₂₇ (p^∗)⁰

we have by Hölder’s inequality

Z

B f(x,u⁺_n)(u_n−u)dx 6 ^ε

3K₁^p∗ Z

B

(u⁺_n)^p∗−1|u_n−u|dx+c₂₇ku_n−uk₁

6 ^ε

3K₁^p∗ku⁺_nk^p_p^∗∗⁻¹ku_n−uk_p^∗+c₂₇|B|^(p1∗)0ku_n−uk_p^∗ 6 ^2ε

3 + ^ε 3 =ε.

So, the sequence (f(·,u⁺_n)(u_n−u))is uniformly integrable inΩ. By Vitali’s theorem we get limn

Z

Ω f(x,u⁺_n)(un−u)dx=0. (3.11) If we choose v=u_n−uin (3.3), pass to the limit asn→∞, and use (3.10) and (3.11), we now get

lim sup

n

hA(un),un−ui=0.

By the(S)₊-property of Awe haveu_n→u inW^1,p(_Ω), which concludes the proof.

The following lemmas show that forλ >0 small enough ˆϕ⁺_λ exhibits the ‘mountain pass’

geometry.

Lemma 3.2. IfHhold, then there existsλ^∗ >0s.t. for allλ∈(0,λ^∗)there existsρ_λ >0s.t.

kuinfk=ρλ

ˆ

ϕ⁺_λ(u) =mˆ⁺_λ >0.

Proof. ByH_g(ii) (iv), for allε>0 we can findc₂₈ =c₂₈(ε)>0 s.t. for a.a.x ∈_{Ω, all}t >⁰ G(x,t)6 ^ε

pt^p+c28t^q

(recall thatq< p). ByH_f (iv)we have as well for a.a.x∈_{Ω, all}t>⁰ F(x,t)6 ^c11

p^∗t^p∗+ ^c11

r t^r− ^c12 p t^p.

Recalling that q<r < p< p^∗ and choosing ε<c₁₂/λwe get for a.a.x∈ _{Ω, all}t>⁰ λG(x,t) +F(x,t)6^λc28t^q+c₂₉t^p∗− ^c30

p t^p,

where, taking c28,c29 > 0 big enough, we may assume c30 > kξk_∞. So, recalling H_β and µ>kξk_∞, for allu∈W^1,p(_Ω)we have

ˆ

ϕ⁺_λ(u)> ^c2

p(p−1)k∇u⁻k^p_p+ ¹ p

Z

Ω(ξ(x) +µ)(u⁻)^pdx + ^c2

p(p−1)k∇u⁺k^p_p+ ¹ p

Z

Ωξ(x)(u⁺)^pdx−

Z

Ω

λG(x,u⁺) +F(x,u⁺)dx

>^c31ku⁻k^p+ ^c2

p(p−1)k∇u⁺k^p_p+ ¹ p

Z

Ω(ξ(x) +c₃₀)(u⁺)^pdx−λc₂₈ku⁺k^q_q−c₂₉ku⁺k^p_p^∗∗

>^c31ku⁻k^p+c₃₂ku⁺k^p−λc₃₃kuk^q−c₃₄kuk^p∗

>^c35kuk^p−λc₃₃kuk^q−c₃₄kuk^p∗ (c₃₁, . . . ,c₃₅>0).

Summarizing, we have

ˆ

ϕ⁺_λ(u)>(c35−jλ(kuk))kuk^p, (3.12) where we have set for alltρ >0

j_λ(ρ) =λc₃₃ρ^q−p+c₃₄ρ^p

∗−p. Sinceq< p< p^∗, we have for allλ>0

lim

ρ→0⁺j_λ(ρ) = lim

ρ→+_∞j_λ(ρ) = +_∞, so there existsρ_λ >0 s.t.

j_λ(ρ_λ) = _inf

ρ>0j_λ(ρ)_. In particular we have

0= j⁰_λ(ρ_λ) =λc₃₃(q−p)ρ^q_λ^−p−1+c₃₄(p^∗−p)ρ^p

∗−p−1

λ ,

which yields

ρ_λ =

λc₃₃(p−q) c₃₄(p^∗−p)

_{p∗ −}¹_q . We are interested in the mappingλ7→ j_λ(ρ_λ), which amounts to

j_λ(ρ_λ) =c₃₆λ

p∗ −p

p∗ −q (c₃₆>0 independent ofλ),

and the latter tends to 0 as λ → 0⁺. So there exists λ^∗ > 0 s.t. for all λ ∈ (0,λ^∗) we have j_λ(ρ_λ)<c₃₅. Thus, by (3.12) we have for allu ∈W^1,p(_Ω)with kuk=ρ_λ

ˆ

ϕ⁺_λ(u)>(c₃₅−j_λ(ρ_λ))ρ^p_λ =: ˆm⁺_λ >0, which concludes the proof.

Let ˆu₁ = uˆ₁(p,ξ₀,β₀) ∈ D+ be the positive, L^p(_Ω)-normalized first eigenfunction of the eigenvalue problem (2.2).

Lemma 3.3. IfHhold, then for allλ>0

t→+lim_∞ϕˆ⁺_λ(tuˆ₁) =−_∞.

Proof. Fix λ > 0. By Hg (ii), H_f (iii), for all η > 0 we can find c37 = c37(η) > 0 s.t. for a.a.

x∈ _{Ω, all}t>^c37

λG(x,t) +F(x,t)> ^η−µ p t^p.

Since ˆu₁ ∈D+, for allt>0 big enough we havetuˆ₁(x)>^c37for allx∈Ω. Then by Lemma2.1 (iv)we have

ˆ ϕ⁺_λ(tuˆ₁)

6^c8

Z

Ω(1+t^p|∇uˆ₁|^p)dx+ ¹ p

Z

Ω(ξ(x) +µ)(tuˆ₁)^pdx+ ¹ p

Z

∂Ωβ(x)(tuˆ₁)^pdσ−

Z

Ω

η

p(tuˆ₁)^pdx 6^c38+

c₃₉kuˆ₁k^p−^η pkuˆ₁k^p_p

t^p (c₃₈,c₃₉ >0).

Choosing η > 0 big enough, the latter tends to −_∞ as soon as t → +∞, concluding the proof.