and Nikolaos S. Papageorgiou

Nonlinear nonhomogeneous Neumann eigenvalue problems

Pasquale Candito

, Roberto Livrea

and Nikolaos S. Papageorgiou

1Department DICEAM, University of Reggio Calabria, 89100 - Reggio Calabria, Italy

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

Received 8 April 2015, appeared 28 July 2015 Communicated by Gabriele Bonanno

Abstract. We consider a nonlinear parametric Neumann problem driven by a nonho- mogeneous differential operator with a reaction which is (p−1)-superlinear near±∞ and exhibits concave terms near zero. We show that for all small values of the parame- ter, the problem has at least five solutions, four of constant sign and the fifth nodal. We also show the existence of extremal constant sign solutions.

Keywords: superlinear reaction, concave terms, maximum principle, extremal constant sign solutions, nodal solution, critical groups.

2010 Mathematics Subject Classification: 35J20, 35J60, 35J92, 58E05.

1 Introduction

Let Ω∈_R^N be a bounded domain with aC²-boundary∂Ω. The aim of this work is to study the existence and multiplicity of solutions with a precise sign information, for the following nonlinear nonhomogeneous parametric (eigenvalue) Neumann problem:

−_diva(Du(z)) = f(z,u(z)_,λ) _inΩ, ^∂u

∂n =_{0 on} ∂Ω. (P_λ) Here n(·) stands for the outward unit normal on ∂Ω. Also, a: R^N → _R^N is a continu- ous and strictly monotone map which satisfies certain other regularity conditions listed in hypothesesH(a)below. These hypotheses are general enough to incorporate as a special case several differential operators of interest, such as the p-Laplacian (1 < p < _{∞), the} (p,q)_- Laplacian (that is, the sum of a p-Laplacian and a q-Laplacian with 1 < q < p < _{∞) and} the generalized p-mean curvature differential operator. The variable λ > 0 is a parameter (eigenvalue) which in general enters in the equation in a nonlinear fashion. The nonlinear- ity of the right-hand side (the reaction of the problem) f(z,x,λ) is a Carathéodory function in (z,x) ∈ _Ω×_R (that is for all x ∈ _R, λ > 0, z 7→ f(z,x,λ) is measurable and for a.a.

z ∈ _{Ω, all} λ > 0, x 7→ f(z,x,λ) is continuous). We assume that x 7→ f(z,x,λ) exhibits (p−1)-superlinear growth near ±∞, while near zero we assume the presence of a concave

BCorresponding author. Email: roberto.livrea@unirc.it

term (that is, of a(p−1)-sublinear term). So, in the reaction f(z,x,λ), we can have the com- peting effects of two different kinds of nonlinearities (“concave–convex” nonlinearities). Such problems were first investigated by Ambrosetti–Brezis–Cerami [2] who deal with semilinear (that is, p = 2) equations. Their work was extended to equations driven by the Dirichlet p-Laplacian, by García Azorero–Manfredi–Peral Alonso [9] and by Guo–Zhang [13]. In all three works the reaction has the special form

f(z,x,λ) =λ|x|^q−2x+|x|^p−2x with 1<q< p< p^∗= ( _{N p}

N−p if p< N, +_∞ if p≥ N.

More general reactions were considered by Hu–Papageorgiou [14] and by Marano–Papa- georgiou [18]. Both papers deal with Dirichlet problems driven by the p-Laplacian. For the Neumann problem, we mention the work of Papageorgiou–Smyrlis [24], where the differential operator is

u7→ −_∆pu+β(z)u for allu∈W^1,p(_Ω) (1< p<_∞), with∆p being thep-Laplace differential operator defined by

∆pu=div(|∇u|^p−2∇u) for allu∈W^1,p(_Ω)

andβ∈ L^∞(_Ω)_, β≥_0, β6= 0. So, in this case the differential operator is coercive. This is not the case in problem (P_λ). Moreover, the reaction in [24] has the form

f(z,x,λ) =λ|x|^q−2x+g(z,x)

with 1< q< pandg(z,x)is a Carathéodory function which is (p−1)-superlinear in x∈ _R. Papageorgiou–Smyrlis [24] look for positive solutions and they prove a bifurcation-type theo- rem describing the set of positive solutions as the parameterλ>0 varies.

Our approach is variational based on the critical point theory. We also use suitable trun- cation and perturbation techniques and Morse theory (critical groups).

2 Mathematical background – hypotheses

In this section, we present the main mathematical tools which we will use in the sequel and state the hypotheses on the data of problem (P_λ). We also present some straightforward but useful consequences of the hypotheses.

Let X be a Banach space and X^∗ its topological dual. By h·_,·i we denote the duality brackets for the pair(X^∗,X). Let ϕ∈ C¹(X). We say thatϕsatisfies the Cerami condition (the C-condition for short), if the following is true:

Every sequence{xn}_n≥1⊆ X s.t. {ϕ(xn)}_n≥1⊆_Ris bounded and (1+kx_nk)ϕ⁰(x_n)→0 in X^∗ as n→_∞, admits a strongly convergent subsequence.

This is a compactness type condition on the functional ϕ, more general than the Palais–

Smale condition. It compensates for the fact that the ambient space X need not be locally compact (being in general infinite dimensional). The C-condition suffices to prove a deforma- tion theorem and then from it derive the minimax theory for the critical values ofϕ. Prominent in that theory is the so-called “mountain pass theorem” (see [3]).

Theorem 2.1. If ϕ∈C¹(X)satisfies the C-condition, x₀,x₁ ∈X withkx₁−x₀k> ρ>0 max{ϕ(x₀)_,ϕ(x₁)}<_inf{ϕ(x)_:kx−x₀k=ρ}=m_ρ

and c = infγ∈_Γmax₀≤t≤1ϕ(γ(t)), where Γ = {γ ∈ C([0, 1],X) : γ(0) = x0, γ(1) = x₁}, then c≥mρ and c is a critical value of ϕ.

The analysis of problem (P_λ), in addition to the Sobolev spaceW^1,p(_Ω), will also involve the Banach spaceC¹(_Ω^¯). This is an ordered Banach space with positive cone

C+ ={u∈C¹(_Ω^¯):u(z)≥0 for allz∈ _Ω^¯}. This cone has a nonempty interior given by

intC+={u∈C¹(_Ω^¯):u(z)>0 for allz∈ _Ω^¯}. Now, let us introduce the hypotheses on the mapa(·).

Letξ ∈C¹(0,∞)with ξ(t)>0 for all t>0 and assume that 0<cˆ≤ ^tξ

0(t)

ξ(t) ≤c0 and c₁t^p−1≤ξ(t)≤c2(1+t^p−1), (2.1) for all t>0, withc₁ >0.

The hypotheses on the mapa(·)are the following:

H(a): a(y) =a₀(|y|)y for all y∈_R^N with a₀(t)>0for all t >0and

(i) a₀ ∈C¹(0,∞), t 7→ta₀(t)is strictly increasing on(0,∞), ta₀(t)→0⁺as t→0⁺and

tlim→0⁺

ta⁰₀(t)

a0(t) = c>−_1;

(ii) k∇a(y)k ≤c3ξ(|y|)

|y| for all y∈_R^N\ {0}and some c3>0;

(iii) (∇a(y)h,h)_RN ≥ ^ξ(|_|^y|)

y| |h|²for all y∈_R^N\ {₀}, all h∈_R^N; (iv) if G₀(t) =Rt

0 sa₀(s)ds for all t ≥ 0, then pG₀(t)−t²a₀(t)≥ −γfor all t ≥0and some γ>0;

(v) there existsτ∈(1,p)such that t 7→G₀(t^1/τ)is convex,lim_t→0^{+ G0}_tτ^(t) =0and t²a₀(t)−τG₀(t)≥ct˜ ^p

for all t>_{0an some}c˜>_0.

Remark 2.2. EvidentlyG₀ is strictly convex and strictly increasing. We setG(y) =G₀(|y|)_for ally∈_R^N. ThenGis convex and it is differentiable at everyy∈_R^N\ {0}. Also

∇G(y) =G₀⁰(|y|) ^y

|y| =a₀(|y|)y =a(y) for ally∈_R^N\ {0}, 0∈ ∂G(0), implies that Gis the primitive of the mapa.

The convexity ofGand the fact thatG(0) =0, imply

G(y)≤(a(y),y)_RN =a₀(|y|)|y|² (2.2) for all y∈_R^N_.

The next lemma is a straightforward consequence of the above hypotheses and summarizes the main properties of the mapa, which we will use in the sequel.

Lemma 2.3. If hypotheses H(a)hold, then

(a) y7→ a(y)is continuous and strictly monotone, hence maximal monotone too;

(b) |a(y)| ≤c₄(1+|y|^p−1)for all y∈_R^N and some c₄>0;

(c) (a(y),y)_RN ≥ _p^c₋¹₁|y|^p for all y∈_R^N.

Lemma2.3and (2.1), (2.2), lead to the following growth estimates for the primitive G.

Corollary 2.4. If hypotheses H(a)hold, then c₁

p(p−1)|y|^p ≤G(y)≤ c₅(1+|y|^p) for all y∈_R^N with c₅ >0.

Example 2.5. The following maps satisfy hypothesesH(a):

(a) a(y) =|y|^p−2ywith 1< p<∞. This map corresponds to the p-Laplacian

∆pu=div(|∇u|^p−2∇u) for all u∈W^1,p(_Ω).

(b) a(y) =|y|^p−2y+µ|y|^q−2ywith 1 <q< pandµ>0. This map corresponds to a sum of a p-Laplacian and aq-Laplacian, that is:

∆pu+µ∆qu for allu∈W^1,p(_Ω).

Such differential operators arise in many physical applications (see [23] and the refer- ences therein).

(c) a(y) = (₁+|y|²)^p−22y with 1< p<∞. This map corresponds to the generalized p-mean curvature differential operator

divh

(1+|∇u|²)^p−22∇ui

for allu∈W^1,p(_Ω). (d) a(y) =|y|^p−2y+ ^|y|p−2y

1+|y|^p with 1< p< _∞.

We introduce the following nonlinear map A: W^1,p(_Ω)→W^1,p(_Ω)^∗ defined by hA(u),yi=

Z

Ω(a(∇u),∇y)_RNdz (2.3) for allu,y∈W^1,p(_Ω)_.

The next result is a particular case of a more general result proved by Gasinski–Papa- georgiou [11].

Proposition 2.6. If hypotheses H(a)hold, then the map A:W^1,p(_Ω)→ W^1,p(_Ω)^∗ defined by(2.3) is bounded (that is, it maps bounded sets to bounded sets), demicontinuous, strictly monotone (hence maximal monotone too) and of type (S)+, that is

u_n *u in W^1,p(_Ω) and lim sup

n→_∞

hA(u_n),u_n−ui ≤0⇒u_n→u in W^1,p(_Ω).

Consider a Carathéodory function f₀: Ω×_R→_Rsatisfying

|f₀(z,x)| ≤α(z)(1+|x|^r−1) for a.a.z∈_Ω, all x∈_R, with α∈ L^∞(_Ω)_{+, 1}< r < p^∗. We set F₀(z,x) =Rx

0 f₀(z,s)ds and consider theC¹-functional ϕ₀: W^1,p(_Ω)→_Rdefined by

ϕ₀(u) =

Z

ΩG(∇u(z))dz−

Z

ΩF₀(z,u(z))dz for allu∈W^1,p(_Ω). In the sequel byk · k_1,p we denote the norm of the Sobolev spaceW^1,p(_Ω), that is

kuk_1,p=kuk^p_p+k∇uk^p_p^1/p. The following result is due to Motreanu–Papageorgiou [21].

Proposition 2.7. If u₀ ∈W^1,p(_Ω)is a local C¹(_Ω^¯)-minimizer ofϕ₀, that is there existsρ₀ >0such that

ϕ₀(u₀)≤ ϕ₀(u₀+h) for all h∈ C¹(_Ω^¯)withkhk_C1(_Ω^¯) ≤ρ₀,

then u0 ∈ C^1,β(_Ω^¯) for some β ∈ (0, 1)and it is also a local W^1,p(_Ω)-minimizer of ϕ0, that is there existsρ₁>0such that

ϕ₀(u₀)≤ ϕ₀(u₀+h) for all h∈W^1,p(_Ω^¯)withkhk_1,p ≤ρ₁.

Remark 2.8. The first such result relating local minimizers, was proved by Brezis–Nirenberg [4], for the spacesC₀¹(_Ω^¯) = {u ∈C¹(_Ω^¯):u_|∂Ω =0}and H¹₀(_Ω)and with G(y) = ¹₂|y|² for all y∈_R^N (it corresponds to the Dirichlet Laplacian).

Now letX be a Banach space andY₂ ⊆ Y₁ ⊆ X. For every integerk ≥0 by H_k(Y₁,Y₂)we denote the kth-singular homology group with integer coefficients for the pair(Y₁,Y2). Recall that H_k(Y₁,Y₂) =0 for all k<0.

Given ϕ∈C¹(X)_andc∈ R, we introduce the following sets:

ϕ^c ={x∈ X: ϕ(x)≤c}, Kϕ ={x ∈X: ϕ⁰(x) =0}, K^c_ϕ ={x∈ Kϕ : ϕ(x) =c}. The critical groups ofϕat an isolated critical pointx₀ ∈Xwithϕ(x₀) =c(that is,x₀∈K^c_ϕ) are defined by

C_k(ϕ,x₀) =H_k(ϕ^c∩ U,ϕ^c∩ U \ {x₀}) for all k≥0,

where U is a neighborhood ofx₀∈ Xsuch thatK_ϕ∩ϕ^c∩ U = {x₀}. The excision property of singular homology implies that the above definition of critical groups is independent of the choice of the neighborhood U.

Next we introduce the hypotheses on the reaction f(z,x,λ)_.

H(f): f: Ω×_R×(0,+_∞) → _R is a function such that for every λ > 0, (z,x) 7→ f(z,x,λ) is Carathéodory, f(z, 0,λ) =0for a.a. z∈ _Ωand

(i) for every ρ > 0and λ > 0, there exists α_ρ(λ) ∈ L^∞(_Ω)₊ such that λ 7→ kα_ρ(λ)k_∞ is bounded on bounded sets and

|f(z,x,λ)| ≤αρ(λ)(z) for a.a.z∈ _Ω, all|x| ≤ρ;

(ii) if F(z,x,λ) =Rx

0 f(z,s,λ)ds, then for allλ>0

x→±lim_∞

F(z,x,λ)

|x|^p = +_∞ uniformly for a.a.z∈ _Ω

and there exist r(λ)∈(p,p^∗)with r(λ)→r₀∈ (p,p^∗)asλ→0⁺andηˆ_∞(λ),η_∞(λ)∈ L^∞(_Ω)_{such that}

ˆ

η_∞(λ)(z)≤lim inf

x→±_∞

f(z,x,λ)

|x|^r(λ)−2x ≤lim sup

x→±_∞

f(z,x,λ)

|x|^r(λ)−2x ≤ η_∞(λ)(z)

uniformly for a.a. z∈ _Ωandλ7→ kηˆ_∞(λ)k_∞,kη_∞(λ)k_∞ are bounded on bounded sets in (_0,+_∞);

(iii) for everyλ>0, there existsθ(λ)∈ max

(r(λ)−p)^N_p, 1 ,p^∗

, andβ₀(λ)such that 0< β₀(λ)≤lim inf

x→±_∞

f(z,x,λ)x−pF(z,x,λ)

|x|^θ(λ) uniformly for a.a.z ∈_Ω;

(iv) for every λ > 0, there exist q(λ),µ(λ) ∈ (1,τ)(see hypotheses H(a) (v)) with q(λ) ≤ µ(λ)andδ₀(λ)∈(0, 1),cˆ₀(λ)>0such that q(λ)→q₀∈ (1,p)asλ→0⁺and

cˆ0(λ)|x|^µ(λ)≤ f(_z,x,λ)_x≤q(λ)_F(_z,x,λ) _{for a.a.z} ∈_Ω, all |x| ≤δ₀(λ)_, there existβ(λ), β₁,β₂>0, withβ(λ)→0⁺asλ→0⁺such that

f(z,x,λ)x≤ β(λ)|x|^q(λ)+β₁|x|^r∗−β₂|x|^p for a.a.z∈_Ω, all x∈_R, with r(λ)≤r^∗ < p^∗ (see (ii)) and there exits a functionη₀(·,λ)∈ L^∞(_Ω)₊such that

kη₀(·,λ)k_∞ →0 asλ→0⁺, lim sup

x→0

F(z,x,λ)

|x|^q(λ) ≤ η0(z,λ) uniformly for a.a.z∈_Ω.

Remark 2.9. Hypotheses H(f) (ii), (iii) imply that for a.a. z ∈ _Ω and all λ > 0, the reac- tion f(z,·,λ)is (p−1)-superlinear near ±∞. Usually such problems are studied using the Ambrosetti–Rabinowitz condition (see [3]). Our hypothesis here is more general and incor- porates in our framework superlinear functions with “slower” growth near±_∞, which fail to satisfy the Ambrosetti–Rabinowitz condition (see the examples below). On this issue, see also [16], [19] and the references therein.

Example 2.10. The following functions satisfy hypotheses H(f). For the sake of simplicity we drop thez-dependence:

f₁(x,λ) =λ|x|^q−2x+|x|^r−2x− |x|^p−2x with 1<q<τ< p<r < p^∗,

f₂(x,λ) =











|x|^r−2x− |x|^p−2x−σ(λ) ifx< −ρ(λ)

λ|x|^q−2x− |x|^p−2x if −ρ(λ)≤ x≤ρ(λ)

|x|^r−2x− |x|^p−2x+σ(λ) ifρ(λ)< x

with 1 < q < τ < p < r < p^∗, ρ(λ) ∈ (0, 1], ρ(λ) → 0⁺ as λ → 0⁺, σ(λ) = (λ−ρ(λ)^r−q)ρ(λ)^q−1_;

f₃(x,λ) =λ|x|^q−2x+|x|^p−2x

ln|x| − ^p−1 p

with 1<q< τ< p.

Note that f₄(·_,λ)does not satisfy the Ambrosetti–Rabinowitz condition.

We introduce the following truncations-perturbations of the reaction f(z,x,λ): fˆ+(z,x,λ) =

(0 ifx ≤0

f(z,x,λ) +β₂x^p−1 if 0<x (2.4) and

fˆ−(z,x,λ) =

(f(z,x,λ) +β₂|x|^p−2x if x<0

0 if 0≤ x. (2.5)

Both are Carathéodory functions. We set Fˆ±(z,x,λ) =

Z _x

0

fˆ±(z,s,λ)ds and introduce theC¹-functionals ˆϕ^λ_±: W^1,p(_Ω)→_Rdefined by

ˆ

ϕ^λ_±(u) =

Z

ΩG(∇u(z))dz+ ^β2

p kuk^p_p−

Z

Ω

Fˆ±(z,u(z),λ)dz for allu∈W^1,p(_Ω). Also, by ϕ_λ :W^1,p(_Ω)→_Rwe denote the energy functional for problem (P_λ) defined by

ϕ_λ(u) =

Z

ΩG(∇u(z))dz−

Z

ΩF(z,u(z),λ)dz for allu∈W^1,p(_Ω). Clearly ϕ_λ ∈C¹(W^1,p(_Ω)).

We conclude this section by fixing our notation. For x ∈ _R let x^± = max{±x, 0}. Then, givenu∈W^1,p(_Ω), we setu^±(·) =u(·)^±. We have

u=u⁺−u⁻, |u|=u⁺+u⁻ and u⁺,u⁻,|u| ∈W^1,p(_Ω).

Giveh(z,x)a jointly measurable function (for example, a Carathéodory function), we define N_h(u)(·) =h(·,u(·)) for all u∈W^1,p(_Ω).

Finally by| · |_N we denote the Lebesgue measure onR^N.

3 Solutions of constant sign

In this section we show that forλ>0 small, problem (P_λ) has at least four nontrivial solutions of constat sign (two positive and two negative).

First we establish the compactness properties of the functionals ˆϕ^λ_± andϕ_λ.

Proposition 3.1. If hypotheses H(a)and H(f)hold andλ > 0, then the functionals ϕˆ^λ_± satisfy the C-condition.

Proof. We do the proof for the functional ˆϕ^λ₊, the proof for ˆϕ^λ₋being similar. So, we consider a sequence{un}inW^1,p(_Ω)such that

|ϕˆ^λ₊(un)| ≤ M₁ (3.1)

for some M₁ >0, alln≥1,

(₁+ku_nk_1,p)(ϕˆ^λ₊)⁰(u_n)→_{0 in}W^1,p(_Ω)^∗ _asn→_{∞. (3.2)}

From (3.2) we have

|h(ϕˆ^λ₊)⁰(un),hi| ≤ ^εnkhk_1,p 1+kunk_1,p for allh∈W^1,p(_Ω)withε_n →0⁺. Hence

hA(u_n)_,hi+β₂ Z

Ω|u_n|^p−2u_nh dz−

Z

Ω

fˆ+(z,u_n,λ)h dz

≤ ^εnkhk_1,p

1+kunk_1,p ^(3.3) for alln≥1. In (3.3) we chooseh=−u⁻_n ∈W^1,p(_Ω). Then, in view of (2.4),

Z

Ω a(−∇u⁻_n),−∇u⁻_n)_RN dz+β₂ku⁻_nk^p_p≤ε_n for alln≥1, so that, because of Lemma2.3,

c₁

p−1k∇u⁻_nk^p_p+β₂ku⁻_nk^p_p ≤ε_n, that is

u⁻_n →0 inW^1,p(_Ω). (3.4)

Next, in (3.3) we chooseh=u⁺_n ∈W^1,p(_Ω). Then

−

Z

Ω a(∇u⁺_n),∇u⁺_n)_RN dz+

Z

Ω f(z,u⁺_n,λ)u⁺_n dz≤εn for alln≥1. (3.5) Also, from (3.1), (3.4) and Corollary2.4, we have

Z

ΩpG(∇u⁺_n)dz−

Z

ΩpF(z,u⁺_n,λ)dz≤ M₂ for some M₂ >0, alln≥1. (3.6) We add (3.5) and (3.6) and obtain

Z

Ω

pG(∇u⁺_n)− a(∇u⁺_n),∇u⁺_n)_RNdz+

Z

Ω

f(z,u⁺_n,λ)−pF(z,un,λ)dz≤ M₃, (3.7) for someM₃ >0, alln≥1. Hence, H(a)(iv) assures that, for alln≥1

Z

Ω

f(z,u⁺_n,λ)−pF(z,u⁺_n,λ)dz≤ M^ˆ₃. (3.8) HypothesesH(f)(i), (iii) imply that we can findb₁(λ)∈(0,β₀(λ))andc₆(λ)>0 such that

b₁(λ)|x|^θ(λ)−c₆(λ)≤ f(z,x,λ)x−pF(z,x,λ) for a.a.z∈ _{Ω, all} x∈_R. (3.9) Using (3.9) in (3.8), we obtain that

{u⁺_n} ⊆L^θ(λ)(_Ω)is bounded. (3.10) Note that in hypothesis H(f) (iii) without any loss of generality, we may assume that 1 ≤ θ(λ)<r(λ). First suppose thatN6= pand lett ∈(0, 1)be such that

1

r(λ) = ¹−t θ(λ)+ ^t

p^∗. (3.11)

The interpolation inequality (see, for example, Gasinski–Papageorgiou [10, p. 905]) implies ku⁺_nk_r(λ) ≤ ku⁺_nk^1−t

θ(λ)ku⁺_nk^t_p∗.

Thus, from (3.10) and the Sobolev embedding theorem ku⁺_nk^r_r⁽₍^λ)

λ)≤ M₄ku⁺_nk^tr_1,p^(λ) for someM₄>0, all n≥1. (3.12) Hypotheses H(f)(i), (ii) imply that we can findc₇(λ)>0 such that

f(z,x,λ)≤c₇(λ)(1+|x|^r(λ)−1) for a.a. z∈_{Ω, all} x∈_R. (3.13) In (3.3) we chooseh= u⁺_n ∈W^1,p(_Ω). Then

Z

Ω a(∇u⁺_n),∇u⁺_n

R^N dz−

Z

Ω f(z,u⁺_n,λ)u⁺_n dz≤ε_n for alln ≥1.

Hence, from Lemma2.3, (3.13) and (3.12), there existc₈(λ),c₉(λ)>0 such that for alln≥1 c₁

p−1k∇u⁺_nk^p_p≤ c₈(λ)(1+ku⁺_nk^r(λ)

r(λ))

≤ c₉(λ)(1+ku⁺_nk^tr_1,p^(λ)). (3.14) Recall that u7→ kuk_θ(λ)+k∇uk_p is an equivalent norm on the spaceW^1,p(_Ω)(see, for exam- ple, [10, p. 227]). Then, (3.10) and (3.14) imply

ku⁺_nk_1,p^p ≤c₁₀(λ)1+ku⁺_nk^tr_1,p^(λ) for somec₁₀(λ)>0, alln≥1. (3.15) From hypothesis H(f) (iii) and after a simple calculation involving (3.11), we show that tr(λ)< p. So, from (3.15) we infer that

{u⁺_n} ⊆W^1,p(_Ω)is bounded. (3.16) If N = p, then by the Sobolev embedding theorem we know thatW^1,p(_Ω),→ L^s(_Ω)continu- ously (in fact compactly) for all s ∈ [1,∞). Then, fors > r(λ) ≥ θ(λ) ≥ 1 sufficiently large, reasoning as in (3.11) and recalling hypothesis H(f)(iii), one has that

tr(λ) = (r(λ)−θ(λ))s s−θ(λ) < p.

Therefore, the previous argument remains valid and so we reach again (3.16).

From (3.4) and (3.16) it follows that

{u_n} ⊆W^1,p(_Ω)is bounded.

At this point, we may assume that there existsu∈W^1,p(_Ω)such that

u_n*u inW^1,p(_Ω) and u_n→u in L^r(λ)(_Ω). (3.17) We return to (3.3), chooseh=un−u and pass to the limit asn→_∞and use (3.17). Then

nlim→_∞hA(u_n)_,u_n−ui=_0,

and Proposition 2.6 implies that u_n → u in W^1,p(_Ω). This proves that the functional ˆϕ^λ₊ satisfies the C-condition.

With minor changes in the proof, we can also have the following result.

Proposition 3.2. If hypotheses H(a)and H(f)hold and λ > 0, then the functional ϕ_λ satisfies the C-condition.

Next we show that for all small values of the parameterλ >0, the functionals ˆϕ^λ_± satisfy the mountain pass geometry (see Theorem2.1).

Proposition 3.3. If hypotheses H(a) and H(f) hold, then there exist λ^∗_± > 0 such that for every λ∈(0,λ^∗_±), we can findρ^λ_±>0for which we have

infh ˆ

ϕ^λ_±(u):kuk_1,p=ρ^λ_± i

= mˆ^λ_±>0.

Proof. By virtue of hypothesis H(f)(iv) we see that given anyλ>0 F(z,x,λ)≤ ^β(λ)

q(λ)|x|^q(λ)+ ^β1

r^∗|x|^r∗− ^β2

p |x|^p for a.a. z∈_Ω, all x∈_R. (3.18) For allu∈W^1,p(_Ω), because of Corollary2.4 and (2.4) we have

ˆ

ϕ^λ₊(u) =

Z

ΩG(∇u)dz+ ^β2

p kuk^p_p−

Z

Ω

Fˆ+(z,u,λ)dz

≥ ^c1

p(p−1)k∇uk^p_p+ ^β2

p kuk^p_p− ^β2

p ku⁺k^p_p−

Z

ΩF(z,u⁺,λ)dz. (3.19) If in (3.19) we use (3.18), we obtain

ˆ

ϕ^λ₊(u)≥ ^c1

p(p−1)k∇uk^p_p+ ^β2

p kuk^p_p− ^β(λ)

q(λ)ku⁺k^q_q⁽₍^λ)

λ)− ^β1 r^∗ku⁺k^r_r^∗∗

≥^hc₁₁−(c₁₂(λ)kuk^q_1,p^(λ)−p+c₁₃kuk^r_1,p^∗−p)ⁱkuk_1,p^p , (3.20) with c₁₁, c₁₃ > 0 independent of λ and c₁₂(λ) → 0 as λ → 0⁺. We introduce the function γ_λ: (0,∞)→(0,∞)defined by

γ_λ(t) =c₁₂(λ)t^q(λ)−p+c₁₃t^r∗−p for allt>0.

Recall that 1<q(λ)< p <r(λ)≤r^∗ < p^∗. Hence

γ_λ(t)→+_{∞ as}t→0⁺and ast→+_∞.

Therefore we can findt₀= t₀(λ)∈(0,∞)such that

γ_λ(t₀) =_min[γ_λ(t)_:t>₀]. In particular,

γ_λ⁰(t₀) = (q(λ)−p)c₁₂(λ)t^q₀^(λ)−p−1+ (r^∗−p)c₁₃t^r₀^∗−p−1=0, hence

t₀= t₀(λ) =

(p−q(λ))c₁₂(λ) (r^∗−p)c₁₃

_{r∗ −}¹_q(

λ)

and a simple calculation leads to

γ_λ(t₀) = [c₁₂(λ)]^q

(λ)−p

r∗ −_q(λ)c₁₄(λ)_,

with λ7→ c₁₄(λ)bounded on bounded intervals. Note that using the hypotheses onq(·)and r^∗, we have

γ_λ(t₀)→0⁺ asλ→0^. So, choosingλ^∗₊ >0 small, we have

γ_λ(t₀)<c₁₁ for allλ∈(0,λ^∗₊). Then, from (3.20) it follows that for allλ∈(0,λ^∗₊)we have

ˆ

ϕ^λ₊(u)≥mˆ^λ₊ >0 for allu∈W^1,p(_Ω) withkuk_1,p =t₀(λ) =ρ^λ₊. In a similar fashion we show the existence ofλ^∗₋>0 such that

ˆ

ϕ^λ₋(u)≥mˆ^λ₋ >0 for allu∈W^1,p(_Ω) withkuk_1,p =t₀(λ) =ρ^λ₋, and the proof is complete.

The next proposition completes the mountain pass geometry for the functionals ˆϕ^λ_±. It is an immediate consequence of the p-superlinear hypothesis H(f)(ii).

Proposition 3.4. If hypotheses H(a)and H(f)hold,λ>₀and u∈intC+, thenϕˆ^λ_±(tu)→ −_∞as t→ ±_∞.

Now we can use variational methods to produce constant sign solutions for problem (P_λ) whenλ>0 is small.

Proposition 3.5. If hypotheses H(a)and H(f)hold, then

(a) for everyλ∈(0,λ^∗₊)problem (P_λ) has at least two positive solutions u₀, ˆu ∈intC+

withu being a local minimizer ofˆ ϕ_λ andϕ_λ(uˆ)<0< ϕ_λ(u₀); (b) for everyλ∈(0,λ^∗₋)problem (Pλ) has at least two negative solutions

v0, ˆv∈ −intC+

withv being a local minimizer ofˆ ϕ_λ andϕ_λ(vˆ)<0< ϕ_λ(v₀);

(c) ifλ^∗ =min{λ^∗₊,λ^∗₋}andλ ∈(0,λ^∗), then problem (P_λ) has at least four nontrivial solutions of constant sign

u₀, ˆu∈intC+, v₀, ˆv∈ −intC+

withu, ˆˆ v local minimizers of ϕ_λandϕ_λ(vˆ),ϕ_λ(uˆ)<0< ϕ_λ(u₀),ϕ_λ(v₀).

Proof. (a) For λ ∈ (0,λ^∗₊), let ρ^λ₊ be as postulated by Proposition 3.3 and consider ¯B_ρλ + = {u ∈ W^1,p(_Ω) : kuk_1,p ≤ ρ^λ₊}, which clearly is weakly compact inW^1,p(_Ω). Moreover, since

ˆ

ϕ^λ₊is sequentially weakly lower semicontinuous inW^1,p(_Ω), one has that there exists ˆu∈ B^¯_ρλ +

such that

ˆ

ϕ^λ₊(uˆ) =infh ˆ

ϕ^λ₊(u):kuk_1,p≤ ρ^λ₊

i≤mˆ^λ₊.

On the other hand, for δ₀ ∈ (0, 1) as in hypothesis H(f)(iv) and ξ ∈ (0,δ₀(λ)) small (take

|ξ|<ρ^λ₊/|_Ω|^1/p_N ), we obtain ˆ

ϕ^λ₊(ξ) =−

Z

ΩF(z,ξ,λ)dz<0.

Therefore, because of Proposition3.3, we can deduce that ub∈ B_ρλ

+ ={u∈W^1,p(_Ω)_:kuk_1,p<ρ^λ₊}_, and

(ϕˆ^λ₊)⁰(uˆ) =0.

So, it follows that

A(uˆ) +β₂|uˆ|^p−2uˆ = N_fˆλ

+(uˆ), (3.21)

where ˆf₊^λ(z,x) = f^ˆ+(z,x,λ). On (3.21) we act with −uˆ⁻ ∈ W^1,p(_Ω) and using (2.4) and Corollary2.4, we obtain ˆu≥0, ˆu6=0. Then, again because of (2.4), (3.21) we have

−diva(∇uˆ(z)) = f(z, ˆu(z),λ) a.e. in Ω, ∂uˆ

∂n =0 on∂Ω,

(see [11]). From [26], we know that ˆu ∈ L^∞(_Ω). So, we can apply the regularity result of Lieberman [17] and infer that ˆu ∈ C+\ {0}. From hypotheses H(f)(i), (iv), we see that for everyλ>0 andρ>0, we can findξ_ρ(λ)>0 such that

f(z,x,λ)x+ξρ(λ)|x|^p ≥0 for a.a. z∈_{Ω, all}|x| ≤ρ.

Letρ=kuˆk_∞ and let ξρ(λ)>0 as above. Then

−diva(∇uˆ(z)) +ξρ(λ)uˆ(z)^p = f(z, ˆu(z),λ) +ξρ(λ)uˆ(z)^p≥0 a.e. in Ω, that is

diva(∇uˆ(z))≤ξρ(λ)uˆ(z)^p a.e. in Ω. (3.22) Letχ(t) =ta₀(t)for allt>0. Then, from H(a)(iii)

tχ⁰(t) =t²a⁰₀(t) +ta₀(t)≥c₁t^p−1, hence, by integration one has

Z _t

0 sχ⁰(s)ds≥ct˜ ^p for allt >0. (3.23) From (3.22), (3.23) and the strong maximum principle of Pucci–Serrin [25, p. 111] we have

ˆ

u(z)>0 for allz∈ _Ω.

So, we can apply the boundary point theorem of Pucci–Serrin [25, p. 120] and have ˆ

u∈intC+. (3.24)

From (2.4) it is clear that

ϕ_λ|C

+ = ϕˆ^λ_+|C

+.

From this equality and (3.24) it follows that ˆu is a local C¹(_Ω^¯)-minimizer of ϕ_λ. Invoking Proposition2.7, we have that ˆuis a localW^1,p(_Ω)-minimizer ofϕ_λ.

Now we look for the second positive solution. Propositions3.1,3.3and3.4permit the use of Theorem2.1on the functional ˆϕ^λ₊. So, we can findu₀∈W^1,p(_Ω)such that

ˆ

ϕ^λ₊(uˆ)<0= ϕˆ^λ₊(0)<mˆ^λ₊ ≤ ϕˆ^λ₊(u₀) and (ϕˆ^λ₊)⁰(u₀) =0. (3.25) From (3.25) it follows that u₀ 6∈ {0, ˆu}, it solves problem (P_λ) and by the nonlinear regularity theory we haveu₀ ∈C+\ {0}(see [17,26]). In fact, as above, using the results of Pucci–Serrin [25, pp. 111, 120], we conclude thatu₀ ∈intC+.

(b) Working in a similar fashion, this time with the function ˆϕ^λ₋, for λ ∈ (0,λ^∗₋)we produce two negative solutions for problem (P_λ)

v₀, ˆv∈ −intC+.

Moreover, ˆvis a local minimizer of ϕ_λ and ϕ_λ(vˆ)<₀< ϕ_λ(v₀)_. (c) Follows from parts (a) and (b).

4 Nodal solutions

In this section, we produce a fifth nontrivial solution of (P_λ), withλ∈(0,λ^∗), which is nodal (sign changing). The idea is first to generate the extremal nontrivial constant sign solutions, that is the smallest nontrivial positive solutionu^∗_λ and the biggest nontrivial negative solution v^∗_λ of (P_λ). Then look for a nontrivial solution in the order interval[v^∗_λ,u^∗_λ] = {u∈ W^1,p(_Ω): v^∗_λ(z) ≤ u(z) ≤ u^∗_λ(z) a.e. in Ω} distinct from v^∗_λ and u^∗_λ. Necessarily, this solution will be nodal.

HypothesesH(f)(i), (ii), (iv) imply that we can findc₁₅ >0 such that

f(z,x,λ)x ≥cˆ₀(λ)|x|^µ(λ)−c₁₅|x|^r(λ) for a.a. z∈ _{Ω, all}x∈_{R, all}λ∈(0,λ^∗). (4.1) This unilateral growth estimate on the reaction f(z,·,λ)leads to the following parametric auxiliary Neumann problem











−diva(∇u(z)) =cˆ₀(λ)|u(z)|^µ(λ)−2u(z)−c₁₅|u(z)|^r(λ)−2u(z) inΩ,

∂u

∂n =0 on∂Ω,

1<µ(λ)< p<r(λ)< p^∗.

(S_λ)

For this auxiliary problem, we have the following existence and uniqueness result for nontrivial solutions of constant sign.

Proposition 4.1. If hypotheses H(a)hold andλ>0, then problem(S_λ)has a unique positive solution u¯_λ ∈ intC+and since problem(S_λ) is oddv¯_λ = −u¯_λ ∈ −intC+ is the unique negative solution of (S_λ).

Proof. First we establish the existence of a positive solution. To this end, letψ⁺_λ : W^1,p(_Ω)→_R be the C¹-functional defined by

ψ⁺_λ(u) =

Z

ΩG(∇u(z))dz+ ¹

pkuk^p− ^cˆ0(λ)

µ(λ)ku⁺k^µ(λ)

µ(λ)+ ^c15

r(λ)ku⁺k^r_r⁽₍^λ)

λ)− ¹

pku⁺k^p_p.

Recall that 1<µ(λ)< p<r(λ)(see (S_λ)). So, using Corollary2.4, we see thatψ⁺_λ is coercive.

Also, using the Sobolev embedding theorem, we can check that ψ_λ⁺ is sequentially weakly lower semicontinuous. Hence, by the Weierstrass theorem, we can find ¯u_λ ∈ W^1,p(_Ω) such that

ψ_λ⁺(u¯λ) =inf{ψ_λ⁺(u):u∈W^1,p(_Ω)}. (4.2) Sinceµ(λ)<r(λ), forξ ∈ (0, 1)small we haveψ⁺_λ(ξ)<0 and so, because of (4.2),

ψ⁺_λ(u¯_λ)<0= ψ_λ⁺(0),

hence ¯u_λ 6=0. From (4.2), we have that ¯u_λ is a critical point ofψ⁺_λ, namely

A(u¯λ) +|u¯λ|^p−2u¯λ= cˆ0(λ)(u¯⁺_λ)^µ(λ)−1−c15(u¯⁺_λ)^r(λ)−1+ (u¯⁺_λ)^p−1_{. (4.3)} On (4.3) we act with −u¯⁻_λ ∈ W^1,p(_Ω) and obtain ¯u_λ ≥ 0, ¯u_λ 6= 0. Hence ¯u_λ is a positive solution of (Sλ). Nonlinear regularity theory implies ¯uλ ∈C+\ {0}. We have

−diva(∇u¯_λ(z)) =cˆ₀(λ)u¯_λ(z)^µ(λ)−1−c₁₅u¯_λ(z)^r(λ)−1 a.e. in Ω, thus

diva(∇u¯_λ(z))≤c₁₅ku¯_λk^r_∞^(λ)−pu¯_λ(z)^p−1 a.e. inΩ, and from [25, p. 111, 120] we conclude that

¯

u_λ ∈intC+.

So, we have established the existence of a positive solution ¯u_λ ∈intC+for problem (S_λ).

Next we show the uniqueness of this positive solution. To this end, consider the integral functionalσ_λ⁺: L¹(_Ω)→_R∪ {+_∞}defined by

σ_λ⁺(_u) = (R

ΩG(∇u^1/τ)dz ifu≥0, u^1/τ ∈W^1,p(_Ω)

+_∞ otherwise.

Let u₁,u₂ ∈ domσ_λ⁺ = {u ∈ L¹(_Ω) : σ_λ⁺(u) < +_∞} (the effective domain of σ_λ⁺) and let t∈ [_{0, 1}]_{. We set}

y= ((1−t)u₁+tu₂)^1/τ, v₁ =u^1/τ₁ , v₂=u^1/τ₂ . From [5, Lemma 1], we have

k∇y(z)k ≤[(1−t)k∇v₁(z)k^τ+tk∇v2(z)k^τ]^1/τ a.e. in Ω, and exploiting the monotonicity ofG₀ and hypothesis H(a)(v)

G(∇y(z)) =G0(k∇y(z)k)≤ G0

((1−t)k∇v₁(z)k^τ+tk∇v2(z)k^τ)^1/τ

≤(1−t)G₀(k∇v₁(z)k) +tG₀(k∇v₂(z)k) for a.a.z∈_{Ω, that is}σ_λ⁺ is convex.

Also, by Fatou’s lemma σ_λ⁺ is lower semicontinuous. Now, letu ∈ W^1,p(_Ω)be a positive solution of problem (S_λ). From the first part of the proof, we haveu∈ _intC+. So, ifh ∈C¹(_Ω^¯) andt ∈(−1, 1)with |t|small, we have

u^τ+th∈_intC+ ∩_domσ_λ⁺.