Positive solutions for ( p, 2 ) -equations with superlinear reaction and a concave boundary term

Positive solutions for ( p, 2 ) -equations with superlinear reaction and a concave boundary term

Nikolaos S. Papageorgiou

and Andrea Scapellato

1National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece

2Università degli Studi di Catania, Dipartimento di Matematica e Informatica, Viale Andrea Doria 6, 95125 Catania, Italy

Received 28 November 2019, appeared 13 January 2020 Communicated by Dimitri Mugnai

Abstract. We consider a nonlinear boundary value problem driven by the (p, 2)- Laplacian, with a(p−1)-superlinear reaction and a parametric concave boundary term (a “concave-convex” problem). Using variational tools (critical point theory) together with truncation and comparison techniques, we prove a bifurcation type theorem de- scribing the changes in the set of positive solutions as the parameter λ>0 varies. We also show that for every admissible parameterλ>0, the problem has a minimal posi- tive solutionu_λand determine the monotonicity and continuity properties of the map λ7→u_λ.

Keywords: concave boundary term, superlinear reaction, (p, 2)-Laplacian, nonlinear regularity, nonlinear maximum principle, positive solutions.

2010 Mathematics Subject Classification: 35J20, 35J60.

1 Introduction

Let Ω ⊆ _R^N be a bounded domain with a C²-boundary ∂Ω. In this paper we study the following nonlinear parametric (p, 2)-equation











−_∆pu(z)−_∆u(z) +ξ(z)u(z)^p−1= f(z,u(z)) inΩ

∂u

∂n_p2 =λu^τ−1 on∂Ω

u>0, λ>0, 1<τ<2< p< N.

, (P_λ)

In this problem,∆p denotes the p-Laplace differential operator defined by

∆pu=div |Du|^p−2Du

for all u∈W^1,p(_Ω), 1< p< N.

The potential function ξ ∈ L^∞(_Ω), ξ(z) ≥ 0 for a.a. z ∈ _Ω, ξ 6≡0. The reaction term f(z,x) is a Carathéodory function (that is, for all x ∈ _R, z 7→ f(z,x) is measurable and for a.a.

BCorresponding author. Email: scapellato@dmi.unict.it

z ∈ _Ω, x 7→ f(z,x) is continuous). We assume that f(z,·) is (p−1)-superlinear satisfying the Ambrosetti–Rabinowitz condition (theAR-conditionfor short). In the boundary condition,

∂u

∂np2 denotes the conormal derivative ofucorresponding to the (p, 2)-Laplace differential op- erator. This directional derivative ofu, is interpreted via the nonlinear Green’s identity (see Papageorgiou–R˘adulescu–Repovš [21], pp. 34, 35). Ifu∈C¹(_Ω), then

∂u

∂np2

=|Du|^p−2+1∂u

∂n

with n(·)being the outward unit normal on ∂Ω. Also λ > 0 is a parameter and τ ∈ (1, 2). So, in problem (P_λ) we have the competing effects of two nonlinearities of different nature.

One is the reaction term which is superlinear (“convex” nonlinearity) and the other is the parametric boundary term, which is sublinear (“concave” nonlinearity). Therefore, problem (P_λ) is a variant of the classical “concave-convex” problem, with the concave term coming from the boundary condition.

The study of “concave-convex” problems was initiated with the seminal paper of Ambrosetti–Brezis–Cerami [2] (semilinear Dirichlet equations). Their work was extended to nonlinear Dirichlet problems driven by the p-Laplacian by García Azorero–Manfredi–Peral Alonso [7] and Guo-Zhang [9]. In these works the reaction has the special form

x7→ λx^τ−1+x^r−1 for allx ≥0, withλ>0 (the parameter) and 1<τ< p<r < p^∗,

p^∗= ( _{N p}

N−p if p< N, +_∞ if N≤ p.

Recently more general reactions and different boundary conditions were considered by Papageorgiou–R˘adulescu–Repovš [18] (semilinear Robin problems), by Leonardi–Papageorgiou [12], Marano–Marino–Papageorgiou [14] (nonlinear Dirichlet problems) and by Papageorgiou–

Scapellato [23] (nonlinear Robin problems). In these works the competition phenomena occur in the reaction of the equation, where we have the presence of concave and convex nonlineari- ties. Problems with parametric concave boundary term were considered by Hu–Papageorgiou [11] (semilinear equations), Papageorgiou–R˘adulescu [16], Papageorgiou–R˘adulescu–Repovš [20], Sabina de Lis–Segura de Leon [25] (nonlinear problems driven by the p-Laplacian). Fi- nally we mention the recent work of Papageorgiou–Scapellato [22] where in the reaction we have the combined effects of linear and superlinear terms.

Our work here extends those of Hu–Papageorgiou [11] and of Sabina de Lis–Segura de Leon [25].

Using variational tools based on the critical point theory, together with truncation and comparison techniques, we prove a bifurcation-type result describing in a precise way the set of positive solutions of problem (P_λ) as the parameterλ > 0 varies. Also we show that for every admissibleλ>0, problem (Pλ) has a smallest positive solution.

We mention that boundary value problems driven by a combination of differential op- erators of different nature (such as (p, 2)-equations), arise in many mathematical models of physical processes. Among the first such examples we mention the Cahn–Hilliard equation (see [4]) describing the process of separation of binary alloys. More recently, we mention the works of Benci–D’Avenia–Fortunato–Pisani [3] (quantum physics) and Cherfils–Il’yasov [5]

(reaction-diffusion systems).

2 Mathematical background – hypotheses

In the study of problem (P_λ), we will use the Sobolev spaceW^1,p(_Ω), the Banach spaceC¹(_Ω) and the boundary Lebesgue spaces L^s(∂Ω)₍₁≤s<_∞).

Byk · kwe denote the norm of the Sobolev spaceW^1,p(_Ω), defined by kuk= kuk^p_p+kDuk^p_p^1p for allu ∈W^1,p(_Ω).

The Banach spaceC¹(_Ω)is an ordered Banach space with positive (order) cone C+ =ⁿu∈ C¹(_Ω):u(z)≥0 for allz∈ _Ω^o.

This cone has a nonempty interior given by

intC+ =u∈C+:u(z)>0 for allz∈_Ω . We will also use another open cone inC¹(_Ω)given by

D+=

u∈C¹(_Ω)_:u(z)>_{0 for all}z∈_Ω, ^∂u

∂n

∂Ω∩u⁻¹(₀) <₀

.

On∂Ωwe consider the(N−1)-dimensional Hausdorff (surface) measureσ(·). Usingσ(·), we can define in the usual way the boundary Lebesgue spacesL^s(∂Ω)(1≤s≤ _∞). We know that there exists a unique continuous linear mapγ0 : W^1,p(_Ω)→ L^p(_∂Ω), known as thetrace map, such that

γ₀(u) =u

∂Ω for allu∈W^1,p(_Ω)∩C(_Ω).

So, the trace map extends the notion of boundary values to all Sobolev functions. This map is compact into L^s(_∂Ω)for all 1≤ s < ^(N_N⁻₋¹_p^)p when p < Nand into L^s(_Ω)for all 1 ≤s <_∞ whenN ≤ p. Moreover, we have

imγ0=W^p10,p(_∂Ω)

1 p+ ¹

p⁰ =1

, kerγ0 =W₀^1,p(_Ω).

In what follows for the sake of notational simplicity we drop the use of the trace map. All restrictions of Sobolev functions on ∂Ωare understood in the sense of traces.

Ifu,v∈W^1,p(_Ω)withu(z)≤v(z)for a.a.z∈ _Ω, then we define

[u,v] ={h ∈W^1,p(_Ω):u(z)≤ h(z)≤ v(z) for a.a. z ∈_Ω}, [u) ={h ∈W^1,p(_Ω):u(z)≤ h(z) for a.a. z∈ _Ω}.

Given g₁,g₂ ∈ L^∞(_Ω), we write g₁ ≺ g₂ if for every K ⊆ _Ω compact we can find c_K > 0 such that

c_K ≤ g₂(z)−g₁(z) _{for a.a.}z∈K.

Note that ifg₁,g₂ ∈C(_Ω)andg₁(z)<g₂(z)for allz∈_{Ω, then} g₁≺ g₂.

We say that a setS ⊆W^1,p(_Ω)_isdownward directed, if givenu₁,u₂ ∈ S, we can find u∈ S such thatu≤u₁,u≤ u₂.

Let h·,·i denote the duality brackets for the pair (W^1,p(_Ω),W^1,p(_Ω)^∗) and let A_p : W^1,p(_Ω)→W^1,p(_Ω)^∗ be the nonlinear operator defined by

hAp(u),hi=

Z

Ω|Du|^p−2(Du,Dh)_RNdz for allu,h∈W^1,p(_Ω).

Proposition 2.1. The operator A_p(·) is bounded (maps bounded sets to bounded sets), continuous, monotone (hence maximal monotone too) and of type(S)+, that is,

u_n −→^w u in W^1,p(_Ω) and lim sup

n→_∞

hA_p(u_n)_,u_n−ui ≤₀ ⇒ u_n→u in W^1,p(_Ω)_. If p=2, then A₂= A∈_L(H¹(_Ω)_,H¹(_Ω)^∗)_.

For x∈_{R, we set}x^±=max{±x, 0}. Then, givenu∈W^1,p(_Ω), we define u^±(z) =u(z)^± for allz∈ _Ω.

We know that

u^± ∈W^1,p(_Ω), u=u⁺−u⁻, |u|= u⁺+u⁻.

Finally, if X is a Banach space andϕ ∈ C¹(X,R), then byK_ϕ we denote the critical set of ϕ(·)_{, that is,}

Kϕ ={u∈W^1,p(_Ω): ϕ⁰(u) =0}. Now we introduce our hypotheses on the data of problem (P_λ).

H(ξ): ξ ∈ L^∞(_Ω),ξ(z)≥0 for a.a. z ∈_Ω,ξ 6≡0.

H(f): f :Ω×_R→_Ris a Carathéodory function such that f(z, 0) =0 for a.a. z ∈_Ωand (i) 0≤ f(z,x)≤ ηx^r−1for a.a. z ∈_Ω, allx≥0, with 0<η, p<r < p^∗;

(ii) ifF(z,x) =Rx

0 f(z,s)ds, then there existϑ₀∈ (p,r)and M>0 such that 0< ϑ₀F(z,x)≤ f(z,x)x for a.a.z∈ _{Ω, all}x≥ M, 0<ess inf

Ω F(·,M).

Remark 2.2. Since we are looking for positive solutions and the above hypotheses concern the positive semiaxisR+= [0,+_∞), without any loss of generality we may assume that

f(z,x) =0 for a.a. z∈_{Ω, all}x≤0. (2.1) Hypothesis H(f)(i)implies that

xlim→0⁺

f(z,x)

x^τ−1 =0 uniformly for a.a.z∈ _Ω. (2.2) Hypothesis H(f)(ii) is the well known AR-condition (unilateral version due to (2.1)). The AR-condition implies that

c₀x^ϑ0 ≤ F(z,x) for a.a. z∈_Ω, allx≥ M, somec₀>0

⇒ c₀x^ϑ0−1≤ f(z,x) for a.a. z∈_{Ω, all}x≥ M

⇒ f(z,·)is(p−1)-superlinear (sinceϑ₀ > p).

It is an interesting open problem whether we can replace the AR-condition by a less re- strictive one as in Papageorgiou–R˘adulescu [17].

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

f₁(x) =

((x⁺)^r−1+ln(1+ (x⁺)^q−1) ifx ≤1

x^s−1 if 1<x with p<r≤ q<_∞, p<s< p^∗, f₂(x) = (x⁺)^r−1 _with p <r< p^∗.

In the sequel, byγ_p:W^1,p(_Ω)→_Rwe denote theC¹-functional defined by γ_p(u) =kDuk^p_p+

Z

Ωξ(z)|u|^pdz for all u∈W^1,p(_Ω).

On account of hypothesis H(ξ) and Lemma 4.11 of Mugnai–Papageorgiou [15], we have c₁kuk^p≤ γp(u) for allu∈W^1,p(_Ω), somec₁ >0. (2.3)

3 Positive solutions

We introduce the following sets

L ={λ>0 : problem (P_λ) admits a positive solution}, S_λ = set of positive solutions of (P_λ).

Proposition 3.1. If hypothesesH(ξ),H(f)hold, thenL 6=∅^{and S}λ ⊆_intC+for allλ>0.

Proof. For everyλ>0, letϕ_λ :W^1,p(_Ω)→_Rbe theC¹-functional defined by ϕ_λ(u) = ¹

pγ_p(u) +¹

2kDuk²₂−

Z

ΩF(z,u⁺)dz− ^λ τ

Z

∂Ω(u⁺)^τdσ for allu∈W^1,p(_Ω). On account of (2.2) and hypothesis H(f)(i), we see that given e > 0, we can find c₂ =c₂(e)>0 such that

F(z,x)≤ex^τ+c₂|x|^r for a.a. z∈ _{Ω, all}x ∈_R.

Then we have ϕ_λ(u)≥ ^c1

pkuk^p−c3[ekuk^τ+kuk^r+λkuk^τ] for somec3>0, allu∈W^1,p(_Ω). (3.1) Here we used (2.3) and the fact that via the trace map the Sobolev space W^1,p(_Ω) is embedded continuously (in fact compactly) intoL^τ(_∂Ω).

Forρ>0, we lete= ¹₂^c_p^1ρp_c^−τ

3 . Then we have c₁

pρ^p−τ−ec₃

ρ^τ = ¹ 2

c₁

pρ^p. (3.2)

Using (3.2) in (3.1) we obtain ϕ_λ(u)≥ ¹

2 c₁

pρ^p−c₃[ρ^r+λρ^τ] _{for all}u∈W^1,p(_Ω)_withkuk= _ρ.

Since p<r, we can chooseρ∈(0, 1)small such that 1

2 c₁

pρ^p−c₃ρ^r≥ η>0.

Then we chooseλ₀ >0 small so that η−λ₀c3ρ^τ ≥ ¹

2η>0

⇒ η−λc₃ρ^τ ≥ ¹

2η>0 for allλ∈(0,λ₀]

⇒ ϕ_λ(u)≥ ¹

2η>0 for allu∈W^1,p(_Ω)withkuk=ρ, all 0<λ≤λ₀. (3.3)

We introduce the open ball

Bρ ={u∈W^1,p(_Ω):kuk<ρ}.

By the Alaoglu and Eberlein-Šmulian theorems, we have that B_ρ is sequentially weakly compact. Also, using the Sobolev embedding theorem and the compactness of the trace map, we see that ϕ_λ(·) is sequentially weakly lower semicontinuous. Invoking the Weierstrass–

Tonelli theorem, we can findu0∈W^1,p(_Ω)_{such that} ϕ_λ(u₀) =min

ϕ_λ(u):u∈B_ρ

. (3.4)

Sinceτ<2< p, we see that

ϕ_λ(u0)<0= ϕ_λ(0)< ¹ 2η

⇒ u₀∈ Bρ\ {0} (see (3.3)). (3.5)

Then from (3.4) and (3.5) it follows that ϕ⁰_λ(u₀) =0,

⇒ hA_p(u₀),hi+hA(u₀),hi+

Z

Ωξ(z)|u₀|^p−2u₀hdz

=

Z

Ω f(x,u⁺₀)hdz+λ Z

∂Ω(u⁺₀)^τ−1hdσ for allh∈W^1,p(_Ω)_{. (3.6)} In (3.6) we chooseh=−u⁻₀ ∈W^1,p(_Ω)_{. Then}

γ_p(u⁻₀) +kDu⁻₀k²₂ =0

⇒ c₁ku⁻₀k^p ≤0 (see (2.3))

⇒ u₀≥0, u₀ 6=0.

From (3.6) we see thatu₀ ∈W^1,p(_Ω)is a positive solution of (Pλ) and we have







−_∆pu₀(z)−_∆u₀(z) +ξ(z)u₀(z)^p−1 = f(z,u₀(z)) for a.a. z∈ _Ω,

∂u₀

∂n_p2 =λu^τ₀⁻¹ on∂Ω. (3.7)

Proposition 2.10 of Papageorgiou–R˘adulescu [17] implies that u₀ ∈ L^∞(_Ω)and then from Theorem 2 of Lieberman [13], we have thatu0 ∈C+\ {0}. From (3.7) we see that

∆pu₀(z) +_∆u0(z)≤ kξk_∞u₀(z)^p−1 for a.a. x∈_Ω

⇒ u₀∈intC+ (see Pucci–Serrin [24], pp. 111, 120).

So, we have proved that

(0,λ₀]⊆_L, that is,L 6=∅^, S_λ ⊆intC+for allλ>0.

Next we show thatL is an interval.

Proposition 3.2. If hypothesesH(ξ),H(f)hold,λ∈_L andµ∈ (_0,λ), thenµ∈_L.

Proof. Since λ ∈ _L, we can find u_λ ∈ S_λ ⊆ intC+ (see Proposition 3.1). We introduce the following truncations of the data of problem (Pµ):

bf(z,x) =

(f(z,x⁺) if x≤u_λ(z)

f(z,u_λ(z)) ifu_λ(z)< x for all(z,x)∈_Ω×_R, (3.8) e_µ(z,x) =

(

µ(x⁺)^τ−1 ifx ≤u_λ(z)

µu_λ(z)^τ−1 ifu_λ(z)<x for all(z,x)∈ _∂Ω×_R. (3.9) Both are Carathéodory functions. We set

Fb(z,x) =

Z _x

0

bf(z,s)ds, E_µ(z,x) =

Z _x

0

e_µ(z,s)ds and consider theC¹-functionalψ_µ :W^1,p(_Ω)→_Rdefined by

ψµ(u) = ¹

pγp(u) +¹

2kDuk²₂−

Z

ΩFb(z,u)dz−

Z

∂ΩEµ(z,u)dσ for all u∈W^1,p(_Ω). From (2.3), (3.8) and (3.9), we see thatψ_µ(·)is coercive. Also it is sequentially weakly lower semicontinuous. Therefore we can findu_µ ∈W^1,p(_Ω)_{such that}

ψµ(uµ) =infh

ψµ(u):u∈W^1,p(_Ω)ⁱ. (3.10) Letu∈intC+ and chooset ∈(0, 1)small (at least so thattu≤u_λ, recall thatu_λ ∈intC+).

Then sinceτ<₂< p, we will have

ψ_µ(tu)<0

⇒ ψ_µ(u_µ)<0=ψ_µ(0) (see (3.10))

⇒ uµ6=0.

From (3.10) we have ψ_µ⁰(u_µ) =0

⇒ hA_p(u_µ),hi+hA(u_µ),hi+

Z

Ωξ(z)|u_µ|^p−2u_µhdz

=

Z

Ω bf(z,u_µ)hdz+

Z

∂Ωe(z,u_µ)hdσ for allh∈W^1,p(_Ω)_{. (3.11)} In (3.11) first we chooseh=−u⁻_µ ∈W^1,p(_Ω). We obtain

γp(u⁻_µ) +kDu⁻_µk²₂=0

⇒ c₁ku⁻_µk^p≤0 (see (2.3))

⇒ u_µ≥0, u_µ6=0.

Next in (3.11) we chooseh= (uµ−u_λ)⁺∈W^1,p(_Ω). We have hA_p(u_µ),(u_µ−u_λ)⁺i+hA(u_µ),(u_µ−u_λ)⁺i+

Z

Ωξ(z)u_µ^p−1(u_µ−u_λ)⁺dz =

=

Z

Ω f(x,u_λ)(u_µ−u_λ)⁺dz+

Z

∂Ωµu^τ_λ⁻¹(u_µ−u_λ)⁺dσ (see (3.8), (3.9))

≤

Z

Ω f(z,u_λ)(uµ−u_λ)⁺dz+

Z

∂Ωλu^τ_λ⁻¹(uµ−u_λ)⁺dz (sinceµ<λ)

= hA_p(u_λ),(u_µ−u_λ)⁺i+hA(u_λ),(u_µ−u_λ)⁺i+

Z

Ωξ(z)u_λ^p−1(u_µ−u_λ)⁺dz (sinceu_λ∈ S_λ)

⇒ u_µ≤u_λ (see Proposition2.1).

So we have proved that

uµ ∈[0,u_λ]\ {0}. (3.12)

From (3.11), (3.12), (3.8), (3.9) it follows that

u_µ ∈S_µ ⊆intC+,

⇒ µ∈_L.

An interesting byproduct of the above proof is the following corollary.

Corollary 3.3. If hypotheses H(ξ), H(f) hold, λ ∈ _L, u_λ ∈ S_λ ⊆ intC+ and µ ∈ (0,λ), then µ∈_L and there exists u_µ ∈S_µ ⊆_intC+such that u_µ≤u_λ.

We can improve this corollary, by imposing an additional mild condition on f(z,·). So, the new hypotheses on the reaction f(z,x)are the following:

H(f)⁰: f : Ω×_R → _R is a Carathéodory function such that f(z, 0) = _{0 for a.a.} z ∈ _Ω, hypotheses H(f)⁰(i),(ii) are the same as the corresponding hypotheses H(f)(i),(ii)and

(i) for every ρ>0, there existsξb_ρ >0 such that for a.a.z∈ _Ωthe function x7→ f(z,x) +ξb_ρx^p−1

is nondecreasing on [0,ρ].

Remark 3.4. The extra condition is a one-sided local Lipschitz condition (recall that p>2). If f(z,·)is differentiable for a.a. z∈_Ωand for everyρ>0, there existsc_ρ>0 such that

f_x⁰(z,x)≥ −cρx^p−2 for a.a. z∈ _{Ω, all}x ∈[0,ρ], then hypothesis H(f)⁰(i)is satisfied.

Proposition 3.5. If hypotheses H(ξ), H(f)⁰ hold, λ ∈ _L, u_λ ∈ S_λ ⊆ _intC+ andµ ∈ (_0,λ), then µ∈_L and we can find uµ ∈Sµ ⊆intC+such that

u_λ−u_µ ∈D+.

Proof. From Corollary 3.3 we already know that µ ∈ _L and we can find uµ ∈ Sµ ⊆ intC+

such that

uµ ≤u_λ. (3.13)

Leta :R^N →_R^N defined by

a(y) =|y|^p−2y+y for all y∈_R^N. Evidentlya ∈C¹(_R^N_,R^N)(recall that p>_{2) and}

∇a(y) =|y|^p−2

I+ (p−2)^y⊗y

|y|²

+I

⇒ (∇a(y)ϑ,ϑ)_RN ≥ |ϑ|² for all y,ϑ∈_R^N. (3.14)

Observe that

diva(Du) =_∆pu+_∆u for allu∈W^1,p(_Ω). (3.15) From (3.13), (3.14), (3.15) and the tangency principle of Pucci-Serrin [24], p. 35, we have

u_µ(z)<u_λ(z) for allz ∈_Ω. (3.16) Let ρ = ku_λk_∞ and letξb_ρ > 0 be as postulated by hypothesis H(f)⁰(i). Let ξe_ρ > ξb_ρ. We have

−_∆pu_µ−_∆u_µ+^hξ(z) +ξe_ρ i

u^p_µ⁻¹

= f(z,uµ) +ξbρu^p_µ⁻¹+^hξeρ−ξbρ

i u_µ^p−1

≤ f(z,u_λ) +ξb_ρu_λ^p−1+^hξe_ρ−ξb_ρ i

u^p_λ⁻¹ (see (3.13) and hypothesis H(f)⁰(i))

= −_∆puλ−_∆uλ+^hξ(z) +ξe_ρ i

u_λ^p−1 for a.a. z∈_Ω. (3.17)

On account of (3.16), we see that h

ξeρ−ξbρ

i

u^p_µ⁻¹ ≺^hξeρ−ξbρ

i u_λ^p−1.

Then from (3.17) and Proposition 3.2 of Gasi ´nski–Papageorgiou [8] we have u_λ−uµ∈ D+.

From Papageorgiou–R˘adulescu–Repovš [19] (see the proof of Proposition 7), we know that S_λ is downward directed. We will use this fact to show that for every λ ∈ _L problem (P_λ) has a smallest positive solutionu_λ ∈S_λ, that is,u_λ≤ ufor allu∈S_λ.

Proposition 3.6. If hypotheses H(ξ), H(f) hold and λ ∈ _L, then problem (P_λ) admits a smallest positive solution

u_λ ∈intC+.

Proof. Since S_λ is downward directed, using Lemma 3.10, p. 178, of Hu–Papageorgiou [10], we can find{u_n}_n≥1⊆S_λ decreasing such that

ninf≥1u_n=infS_λ. We have

hAp(un),hi+hA(un),hi+

Z

Ωξ(z)u^p_n⁻¹hdz=

Z

Ω f(z,un)hdz+λ Z

∂Ωu^τ_n⁻¹hdσ (3.18) for all h∈W^1,p(_Ω), alln∈ _N.

In (3.18) we choose h = u_n ∈ W^1,p(_Ω). Since 0 ≤ u_n ≤ u₁ for all n ∈ _N, using (2.3) and hypothesis H(f)(i), we see that

{u_n}_n≥1⊆W^1,p(_Ω)is bounded.

From Lieberman [13] (Theorem 2), we see that there existα∈ (0, 1)andc₄>0 such that u_n∈C^1,α(_Ω) and ku_nk_C1,α(Ω)≤c₄ for all n∈_N.

Recall that C^1,α(_Ω) ,→ C¹(_Ω) compactly. This fact and the monotonicity of the sequence {un}_n≥1imply that there existsuλ ∈C¹(_Ω)such that

un→u_λ inC¹(_Ω)asn→_∞. (3.19)

We need to show thatu_λ 6= 0. To this end we consider the following auxiliary boundary value problem











−_∆pu(z)−_∆u(z) +ξ(z)u(z)^p−1=0 inΩ

∂u

∂n_p2 = λu^τ−1 on∂Ω

u>0,λ>0, τ<2< p

. (Q_λ)

Claim 1. For everyλ>0problem(Q_λ)admits a unique solutionue_λ ∈intC+.

First we show the existence of a positive solution for problem (Q_λ). For this purpose we introduce theC¹-functionalβ_λ :W^1,p(_Ω)→_Rdefined by

β_λ(u) = ¹

pγ_p(u) + ¹

2kDuk²₂− ^λ τ

Z

∂Ω(u⁺)^τdσ for allu∈W^1,p(_Ω). From (2.3) and sinceτ<2< p, we see that

β_λ(·)is coercive.

Also the Sobolev embedding theorem and the compactness of the trace map, imply that β_λ(·)is sequentially weakly lower semicontinuous.

So, we can findueλ ∈W^1,p(_Ω)such that β_λ(u_eλ) =minh

β_λ(u):u∈W^1,p(_Ω)ⁱ. (3.20) Sinceτ<2< p, we infer that

β_λ(u_eλ)<0= β_λ(0)

⇒ u_eλ 6=0.

From (3.20) we have β⁰_λ(u_eλ) =0

⇒ hA_p(u_eλ),hi+hA(u_eλ),hi+

Z

Ωξ(z)|u_eλ|^p−2u_eλhdz=λ Z

∂Ω(u_e⁺_λ)^τ−1hdσ for allh∈W^1,p(_Ω).

Choosingh=−u_e⁻_λ ∈W^1,p(_Ω)and using (2.3), we infer that ue_λ ≥0, ue_λ 6=0.

Moreover, as before (see the proof of Proposition3.1), using the nonlinear regularity theory of Lieberman [13] (Theorem 2) and the nonlinear maximum principle of Pucci–Serrin [24]

(p. 120), we conclude that

ue_λ ∈_intC+. (3.21)

Now we show the uniqueness of this positive solution of problem (Q_λ). To this end, we consider the integral functional jλ: L¹(_Ω)→_R=_R∪ {+_∞}defined by

j_λ(u) = (₁

pkDu¹²k^p_p+¹₂kDu¹²k²₂+ ¹_pR

Ωξ(z)u^p2 dz, ifu≥0,u¹² ∈W^1,p(_Ω)

+_∞, otherwise.

From Diaz–Saá [6] (Lemma 1), we know that j_λ(·)is convex.

Let domjλ = {u ∈ L¹(_Ω) : jλ(u)< _∞}(the effective domain of jλ(·)). Letveλ be another positive solution of (Q_λ). Reasoning as we did forue_λ, we show that

ve_λ ∈intC+. (3.22)

Then from (3.21), (3.22) and Proposition 4.1.22, p. 274, of Papageorgiou–R˘adulescu–Repovš [21], we have ^ueλ

evλ,^veλ

ueλ ∈L^∞(_Ω). Leth= u_e²_λ−v_e²_λ. For t∈[0, 1]we have ue²_λ−th∈domj_λ and ve²_λ+th∈domj_λ.

Thenj_λ(·)is Gâteaux differentiable atue²_λ and atve²_λ in the directionh. Moreover, using the nonlinear Green’s identity, we have

j⁰_λ(u_e²_λ)(h) = ^λ 2

Z

∂Ωue^τ_λ⁻²(u_e²_λ−v_e²_λ)dσ, j⁰_λ(v_e²_λ)(h) = ^λ

2 Z

∂Ωve^τ_λ⁻²(u_e²_λ−v_e²_λ)dσ.

Sincej_λ(·)is convex, we have thatj⁰_λ(·)is monotone. Sinceτ<2 we have 0≤ ^λ

2 Z

∂Ω

"

1

ue²_λ^−τ − ¹ ve²_λ^−τ

#

(u_e²_λ−v_e²_λ)_dσ ≤₀

⇒ u_eλ = _ev_λ.

Therefore the positive solutionue_λ ∈intC+is unique. This proves Claim 1.

This solution provides a lower bound for the elements ofS_λ. Claim 2.ue_λ≤ u for all u∈ S_λ.

Letu∈S_λ ⊆intC+. We introduce the following Carathéodory function b_λ(z,x) =

(

λ(x⁺)^τ−1 ifx ≤u(z)

λu(z)^τ−1 ifu(z)<x for all(x,z)∈_∂Ω×_R. (3.23) We setB_λ(z,x) = Rx

0 b_λ(z,s)ds and consider theC¹-functional ϑ_λ :W^1,p(_Ω)→_R defined by

ϑ_λ(u) = ¹

pγp(u) + ¹

2kDuk²₂−

Z

∂ΩB_λ(z,u)dσ for all u∈W^1,p(_Ω).

From (3.23) and (2.3) it is clear thatϑ_λ(·)is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find ub_λ ∈W^1,p(_Ω)such that

ϑ_λ(u_bλ) =infh

ϑ_λ(u):u∈W^1,p(_Ω)ⁱ. (3.24)

As before (see Claim 1), sinceτ<2< p, we see that ϑ_λ(u_bλ)<0= ϑ_λ(0)

⇒ u_bλ 6=0.

From (3.24) we have ϑ⁰_λ(u_bλ) =0

⇒ hAp(u_bλ),hi+hA(u_bλ),hi+

Z

Ωξ(z)|u_bλ|^p−2u_bλhdz=

Z

∂Ωb_λ(z,ub_λ)hdσ (3.25) for allh∈W^1,p(_Ω).

As before (see the proof of Proposition3.2), if in (3.25) we choose firsth=−u_e⁻_λ ∈W^1,p(_Ω) and thenh= (u_bλ−u)⁺∈W^1,p(_Ω)and using (3.23), we show that

ub_λ ∈[0,u]\ {0}. (3.26)

From (3.26), (3.23), (3.25) and Claim 1, it follows that ub_λ =u_eλ

⇒ u_eλ ≤u for allu∈Sλ (see (3.26)).

This proves Claim 2.

From (3.19) and Claim 2, we have ue_λ≤ u_λ

⇒ uλ6=0 and souλ∈ Sλ ⊆intC+,uλ =infSλ. Proposition 3.7. If hypothesesH(ξ),H(f)hold and0< µ<λ∈_L, then

(a) uµ ≤uλ; (b) ue_µ ≤u_eλ. Proof.

(a) Let u_λ ∈ intC+ be the minimal positive solution of problem (P_λ) (see Proposition 3.6).

On account of Corollary3.3, we can finduµ ∈Sµ∈intC+such that u_µ≤ u_λ

⇒ u_µ≤ u_λ recall thatu_µ ≤ufor allu ∈S_µ. (b) Let eeµ(z,x)be the Carathéodory function defined by

ee_µ(z,x) = (

µ(x⁺)^τ−1 ifx≤u_eλ(z)

µue_λ(z)^τ−1 ifue_λ(z)<x for all(z,x)∈∂Ω×_R. (3.27) We setEeµ(z,x) =Rx

0 eeµ(z,s)dsand consider theC¹-functionalϕeµ :W^1,p(_Ω)→_Rdefined by

ϕeµ(u) = ¹

pγ_p(u) + ¹

2kDuk²₂−

Z

∂ΩEeµ(z,u)dz for allu∈W^1,p(_Ω).

Evidentlyϕe_µ(·)is coercive (see (3.27) and (2.3)) and sequentially weakly lower semicon- tinuous. So, we can findubµ ∈W^1,p(_Ω)such that

ϕe_µ(u_bµ) =infh

ϕe_µ(u):u∈W^1,p(_Ω)ⁱ<0= ϕ_eµ(0) (sinceτ<2< p)

⇒ u_bµ 6=0.

We have

hϕ_e⁰_µ(u_bµ),hi=0 for all h∈W^1,p(_Ω).

Choosing h=−u_b⁻_µ ∈W^1,p(_Ω)andh= (u_bµ−u_eλ)⁺∈W^1,p(_Ω), we obtain ubµ ∈[0,ue_λ], ubµ 6=0

⇒ u_bµ =u_eµ ∈_intC+ (see (3.27) and Claim 1 in the proof of Proposition3.6)

⇒ u_eµ ≤u_eλ. Let 0< µ < λandη₀ = ^η

µ. Thenη ≤ λη₀. Motivated by hypothesis H(f)(i), we consider the following auxiliary boundary value problem











−_∆pu(z)−_∆u(z) +ξ(z)u(z)^p−1 =λη₀u(z)^r−1 inΩ,

∂u

∂n_p2 =λu^τ−1 on∂Ω,

u>0,λ>0,τ<2< p<r.

(R_λ)

Reasoning as in the proofs of Propositions3.1and3.6, we obtain the following result.

Proposition 3.8. If hypothesis H(ξ) holds andλ ∈ _L, then problem (R_λ) admits a smallest positive solution u^∗_λ ∈intC+ and there exists u_λ ∈S_λ ⊆intC+such that

ue_λ ≤u_λ ≤u^∗_λ. Letλ^∗ =supL.

Proposition 3.9. If hypothesesH(ξ),H(f)hold, thenλ^∗ <_∞.

Proof. Letµ∈(_0,λ)_{and set 0}< m_eµ =_min

Ω ue_µ (recall thatue_µ ∈ _intC+). From Propositions3.8 and3.7(b), we have

0<m_eµ ≤u_eλ ≤u^∗_λ. We have











−_∆pu^∗_λ−_∆u^∗_λ+ξ(z)(u^∗_λ)^p−1 =λη0(u^∗_λ)^r−1 inΩ,

∂u^∗_λ

∂n_p2 = λ(u^∗_λ)^τ−1 on∂Ω, λ>0, τ<2< p< r.

(3.28)

Leta(z) =η₀(u^∗_λ(z))^r−2 andd(z) =u^∗_λ(z)^τ−2. Thena∈ L^∞(_Ω)andd∈ C(_Ω). We rewrite (3.28) using a(·)_andd(·). So, we have











−_∆pu^∗_λ−_∆u^∗_λ+ξ(z)(u^∗_λ)^p−1 =λa(z)u^∗_λ inΩ,

∂u^∗_λ

∂np2

=λd(z)u^∗_λ on∂Ω,

λ>0.

(3.29)

Let

Wbp =

w∈W^1,p(_Ω):k(w) =

Z

Ωa(z)wdz+

Z

∂Ωd(z)wdσ=0

.

We have W^1,p(_Ω) = _R⊕Wb_p (see Abreu-Madeira [1], Lemma 2.2). Then from (3.29) and Theorem 1.1 of [1], we have

0<λ≤_bc inf

"₁

pγ_p(w) +¹₂kDwk²₂

k(w) ^:w∈Wb_p, w6=0

#

<_∞ for somecb>0.

This fact combined with Proposition3.8implies that we have λ^∗ <_∞.

Proposition 3.10. If hypothesesH(ξ),H(f)⁰ hold andλ ∈(0,λ^∗), then problem(P_λ)admits at least two positive solutions:

u₀,ub∈intC+, u₀ ≤u,_b u₀ 6=u._b

Proof. Let ϑ ∈ (λ,λ^∗). Using Proposition 3.5 we can find u₀ ∈ S_λ ⊆ intC+ and u_ϑ ∈ S_ϑ ⊆ intC+such that

u_ϑ−u₀ ∈D+. (3.30)

We introduce the following truncations of the data of (P_λ)

µb(z,x) =

(f(z,u₀(z)) if x≤u₀(z)

f(z,x) ifu0(z)<x for all(z,x)∈_Ω×_R, (3.31) wb_λ(z,x) =

(λu₀(z)^τ−1 if x≤u₀(z)

λx^τ−1 ifu₀(z)<x for all(z,x)∈_∂Ω×_R. (3.32) These are Carathéodory functions. We set

Mb(z,x) =

Z _x

0 µb(z,s)ds and Wb_λ(z,x) =

Z _x

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

db_λ(u) = ¹

pγ_p(u) + ¹

2kDuk²₂−

Z

ΩMb(z,u)dz−

Z

∂ΩWb_λ(z,u)dσ for all u∈W^1,p(_Ω). In addition, we introduce the following truncations ofµb(z,·)and ofwb_λ(z,·)

µb₀(z,x) = (

µb(z,x) if x≤u_ϑ(z)

µb(z,u_ϑ(z)) ifu_ϑ(z)<x for all(z,x)∈_Ω×_{R, (3.33)} wb⁰_λ(z,x) =

(

wb_λ(z,x) ifx≤ u_ϑ(z)

wb_λ(z,u_ϑ(z)) _ifu_ϑ(z)< x for all(z,x)∈_∂Ω×_R. (3.34) These are Carathéodory functions. We set

Mb0(z,x) =

Z _x

0 µb₀(z,s)ds and Wb_λ⁰(z,x) =

Z _x

0 wb⁰_λ(z,s)ds and consider theC¹-functionaldb_λ⁰:W^1,p(_Ω)→_Rdefined by

db_λ⁰(u) = ¹

pγ_p(u) + ¹

2kDuk²₂−

Z

ΩMb₀(z,u)dz−

Z

∂ΩWb_λ⁰(z,u)dσ for allu∈W^1,p(_Ω).

From (3.31), (3.32), (3.33) and (3.34) it is clear that db_λ

[0,uϑ] = db_λ⁰

[0,uϑ] and db_λ⁰

[0,uϑ] = db_λ⁰0

[0,uϑ]. (3.35) Moreover, we have

Kdbλ

⊆[u₀)∩intC+ (see (3.31), (3.32)) (3.36) K_db0

λ

⊆[u0,u_ϑ]∩intC+ (see (3.33), (3.34)). (3.37) From (3.35) and (3.36) we see that without any loss of generality we may assume that

K_db

λ∩[0,uϑ] ={u0}. (3.38)

Otherwise we already have a second positive smooth solution of (P_λ) bigger thanu₀ (see (3.36)) and so we are done.

From (3.33), (3.34) and (2.3) it is clear that db_λ⁰(·)is coercive. Also it is sequentially weakly lower semicontinuous. So, we can findue₀ ∈W^1,p(_Ω)such that

db_λ⁰(u_e0) =minh

db_λ⁰(u):u∈W^1,p(_Ω)ⁱ

⇒ u_e0∈[u₀,u_ϑ]∩intC+ (see (3.37))

⇒ u_e0∈K_db

λ (see (3.35))

⇒ u_e0=u₀ (see (3.38)).

From (3.30) and (3.35) it follows that

u0 is a localC¹(_Ω)-minimizer ofd_λ

⇒ u₀ is a localW^1,p(_Ω)-minimizer ofd_λ

(see Papageorgiou–R˘adulescu [17], Proposition 2.12).

We assume thatK

dbλ is finite or otherwise on account of (3.36) we already have an infinity of positive smooth solutions bigger than u0 and so we are done. Invoking Theorem 5.7.6, p. 449, of Papageorgiou–R˘adulescu–Repovš [21], we can findρ∈ (0, 1)small such that

db_λ(u₀)<inf[d_λ(u):ku−u₀k= ρ] =m_bλ. (3.39) Moreover, on account of hypothesis H(f)⁰(ii)=_H(f)(ii), we have that

db_λ(·)satisfies the Palais–Smale condition (3.40) and ifu∈intC+, then

db_λ(tu)→ −_∞ast →+_∞. (3.41)

Then (3.39), (3.40) and (3.41) permit the use of the mountain pass theorem. So, we can find ub∈W^1,p(_Ω)such that

ub∈K_db

λ and mb_λ ≤d_λ(u_b)

⇒ u₀≤ u_b∈intC+(see (3.36)),u₀6= u_b(see (3.39)),ub∈S_λ (see (3.31), (3.32)).

Proposition 3.11. If hypothesesH(ξ),H(f)hold, thenλ^∗ ∈_L.