Existence results for double phase problems depending on Robin and Steklov eigenvalues for the p-Laplacian

Said El Manouni; Greta Marino; Patrick Winkert

doi:10.1515/anona-2020-0193

Open Access Published by De Gruyter July 29, 2021

Existence results for double phase problems depending on Robin and Steklov eigenvalues for the p-Laplacian

Said El Manouni , Greta Marino and Patrick Winkert

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2020-0193

Abstract

In this paper we study double phase problems with nonlinear boundary condition and gradient dependence. Under quite general assumptions we prove existence results for such problems where the perturbations satisfy a suitable behavior in the origin and at infinity. Our proofs make use of variational tools, truncation techniques and comparison methods. The obtained solutions depend on the first eigenvalues of the Robin and Steklov eigenvalue problems for the p-Laplacian.

Keywords: Convection term; double phase operator; multiplicity results; nonlinear boundary condition; Robin eigenvalue problem; Steklov eigenvalue problem

MSC 2010: 35J15; 35J62; 35J92; 35P30

1 Introduction

Let Ω ⊂ ℝ^N, N > 1, be a bounded domain with Lipschitz boundary ∂Ω. We consider the following double phase problem with nonlinear boundary condition and convection term given by

(1.1) −div(|∇u|p−2∇u+μ(x)|∇u|q−2∇u)=h1(x,u,∇u) in Ω, (|∇u|p−2∇u+μ(x)|∇u|q−2∇u)⋅v=h2(x,u) on ∂Ω,

where ν(x) is the outer unit normal of Ω at the point x ∈ ∂Ω, 1 < p < q < N, 0 ≤ μ(·) ∈ L1(Ω) and h₁ : Ω × R × ℝ^N → R as well as h₂ : ∂Ω × R → R are Carathéodory functions which satisfy suitable structure conditions and behaviors near the origin and at infinity, see Sections 3 and 4 for the precise assumptions.

The differential operator that appears in (1.1) is the so-called double phase operator which is defined by

(1.2) −div(|∇u|p−2∇u+μ(x)|∇u|q−2∇u) for u∈W1,H(Ω)

with an appropriate Musielak-Orlicz Sobolev space W^1,ℌ(Ω), see its definition in Section 2. Note that when inf_Ω μ > 0 or μ ≡ 0 then the operator becomes the weighted (q, p)-Laplacian or the p-Laplacian, respectively.

The energy functional J : W^1,ℌ(Ω) → R related to the double phase operator (1.2) is given by

(1.3) J(u)=∫Ω(|∇u|p+μ(x)|∇u|q)dx,

where the integrand has unbalanced growth. The main characteristic of the functional J is the change of ellipticity on the set where the weight function is zero, that is, on the set {x ∈ Ω : μ(x) = 0}. Precisely, the energy density of J exhibits ellipticity in the gradient of order q on the points x where μ(x) is positive and of order p on the points x where μ(x) vanishes.

The first who introduced and studied functionals whose integrands change their ellipticity according to a point was Zhikov [37] (see also the monograph of Zhikov-Kozlov-Oleinik [38]) in order to provide models for strongly anisotropic materials. Functionals stated in (1.3) have been intensively studied in the past decade concerning regularity for isotropic and anisotropic functionals. We mention the papers of Baroni-Colombo-Mingione [3, 4, 5], Baroni-Kuusi-Mingione [6], Byun-Oh [7], Colombo-Mingione [9, 10], Marcellini [21, 22], Ok [25, 26], Ragusa-Tachikawa [33] and the references therein.

In this paperwe are going to study problem (1.1) concerning multiplicity of solutions. In the first part of the paper, see Section 3,we prove the existence of a nontrivialweak solutionwhen the function h ₁ depends on the gradient of the solution. Hence, no variational tools like critical point theory are available. We will make use of the surjectivity result for pseudomonotone operatorswhere in the proof the first eigenvalues of the Robin and Steklov eigenvalue problems for the p-Laplacian play an important role. In the second part of the paper we will skip the gradient dependence and prove the existence of two constant sign solutions, one is nonnegative and the other one is nonpositive. Here, we need some stronger conditions on the nonlinearities, for example superlinearity at ±∞. Again, the solutions depend on the first Robin and Steklov eigenvalues, respectively. We will see that the Steklov eigenvalue problem is the more natural one for problems with nonlinear boundary condition than the Robin eigenvalue problem.

There are only few works dealing with double phase operators along with a nonlinear boundary condition. Papageorgiou-Vetro-Vetro [29] studied the Robin problem

(1.4) −div(a(z)|∇u|p−2∇u) )−Δqu+ξ(z)|u|p−2u=λf(z,u(z)) in Ω, ∂u∂nθ+β|u|p−2u=0 on ∂Ω,

where 1 < q < p < N, ξ ∈ L^∞(Ω) is a positive potential, a(z) > 0 for a. a. z ∈ Ω and

∂u∂nθ=[ a(z)|∇u|p−2+|∇u|q−2 ]∂u∂n

with n(·) being the outward unit normal on ∂Ω. Under different assumptions it is shown that problem (1.4) admits two nontrivial solutions u_λ , ˆu_λ ∈ W^1,ℌ(Ω) for small λ > 0 such that ‖u_λ‖_1,H → +∞and ‖ˆu_λ‖_1,H → 0 as λ → 0⁺. In Papageorgiou-Rădulescu-Repovš [28] the authors proved the existence of multiple solutions in the superlinear and the resonant case for the problem

−div(a0(z)|∇u|p−2∇u) )−Δqu+ξ(z)|u|p−2u=f(z,u(z)) in Ω, ∂u∂nθ+β|u|p−2u=0 on ∂Ω,

where 1 < q < p ≤ N and with a positive Lipschitz function a₀(·). Note that our assumptions and our treatment differ from the ones in [28] and [29]. Also, we allow that the weight function could be zero at some points. Recently, Gasiński-Winkert [17] considered the problem

(1.5) −div(|∇u|p−2∇u+μ(x)|∇u|q−2∇u)=f(x,u)−|u|p−2u−μ(x)|u|q−2u in Ω, (|∇u|p−2∇u+μ(x)|∇u|q−2∇u)⋅v=g(x,u) on ∂Ω.

Based on the Nehari manifold method it is shown that problem (1.5) has at least three nontrivial solutions. We point out that the proof for the constant sign solutions in [17] is based on a mountain-pass type argument and so different from the treatment we used in this paper. Very recently, Farkas-Fiscella-Winkert [13] studied singular Finsler double phase problems with nonlinear boundary condition and critical growth of the form

(1.6) −div(A(u))+up−1+μ(x)uq−1=up*−1+λ(uγ−1+g1(x,u)) in Ω, A(u)⋅v=up*−1+g2(x,u) on ∂Ω, u>0 in Ω,

where

div(A(u)):=div(Fp−1(∇u)∇F(∇u)+μ(x)Fq−1(∇u)∇F(∇u))

is the so-called Finsler double phase operator and (ℝ^N , F) stands for a Minkowski space. The existence of one weak solution of (1.6) is proven by applying variational tools and truncation techniques.

For existence results for double phase problems with homogeneous Dirichlet boundary condition we refer to the papers of Colasuonno-Squassina [8] (eigenvalue problem for the double phase operator), Farkas-Winkert [12] (Finsler double phase problems), Gasiński-Papageorgiou [14] (locally Lipschitz right-hand side), Gasiński-Winkert [15, 16] (convection and superlinear problems), Liu-Dai [19] (Nehari manifold approach), Marino-Winkert [23] (systems of double phase operators), Perera-Squassina [31] (Morse theoretical approach), Zeng-Bai-Gasiński-Winkert [35, 36] (multivalued obstacle problems) and the references therein. Relatedworks dealing with certain types of double phase problems can be found in the works of Bahrouni-Rădulescu-Winkert [1] (Baouendi-Grushin operator), Barletta-Tornatore [2] (convection problems in Orlicz spaces), Liu-Dai [20] (unbounded domains), Papageorgiou-Rădulescu-Repovš [27] (discontinuity property for the spectrum), Rădulescu [32] (overview about isotropic and anisotropic double phase problems) and Zeng-Bai-Gasiński-Winkert [34] (convergence properties for double phase problems). Finally, we mention the nice overview article of Mingione-Rădulescu [24] about recent developments for problems with nonstandard growth and nonuniform ellipticity.

The paper is organized as follows. In Section 2 we recall the main properties of the double phase operator including the properties of the Musielak-Orlicz Sobolev space W^1,ℌ(Ω). In Section 3 we prove the existence of at least one solution of (1.1) when h₁ depends on the gradient of the solution, see Theorem 3.1. The proof is based on the surjectivity result for pseudomonotone operators and on the properties of the eigenvalues of the Robin and Steklov eigenvalue problems for the p-Laplacian. Finally, in the last section, we skip the convection term and use variational tools in order to prove the existence of two constant sign solutions for superlinear problems. We consider two different problems. The first problem is treated by properties of the first Steklov eigenvalue and the second one by the first Robin eigenvalue, see Theorems 4.1 and 4.2.

2 Preliminaries

In this section we recall some definitions and present the main tools which will be needed in the sequel.

For every 1 ≤ r < ∞ we denote by L^r(Ω) and L^r(Ω;ℝ^N) the usual Lebesgue spaces equipped with the norm · ‖_r and for 1 < r < ∞we consider the corresponding Sobolev space W^1,r(Ω) endowed with the norm ‖·‖_1,r. It is known that W^1,r(Ω) → L^ˆr(Ω) is compact for ˆr < r^* and continuous for ˆr = r^*,where r^* is the critical exponent of r defined by

(2.1) r⋆={ NrN−r if r<N, any l∈(r,∞) if r≥N.

On the boundary ∂Ω of Ω we consider the (N −1)-dimensional Hausdorff (surface) measure σ and denote by L^r(∂Ω) the boundary Lebesgue space with norm ‖·‖_r_,∂Ω. Fromthe definition of the tracemappingwe know that W^1,r(Ω) → L^˜r(∂Ω) is compact for ˜r < r_* and continuous for ˜r = r_*, where r_* is the critical exponent of r on the boundary given by

(2.2) r*={ (N−1)rN−r if r<N, any l∈(r,∞) if r≥N.

For simplification we will avoid the notation of the trace operator throughout the paper.

In the entire paper we will assume that

(2.3) 1<p<q<N and 0≤μ(⋅)∈L1(Ω).

Note that the conditions in (2.3) are quite general. In all the other mentioned works for Neumann double phase problems (see, for example, [13], [17], [28], [29]) the condition

NqN+q−1<p

is needed, which is equivalent to q < p_* and so q < p^* is also satisfied. We do not need this restriction in the current paper.

Let ℌ: Ω × [0, ∞) → [0, ∞) be the function defined by

H(x,t)=tp+μ(x)tq.

Based on this we can introduce the modular function given by

ρH(u):=∫ΩH(x,|u|)dx=∫Ω(|u|p+μ(x)|u|q)dx.

Then, the Musielak-Orlicz space L^ℌ(Ω) is defined by

LH(Ω)={ u∣u:Ω→ℝ is measurable and ρH(u)<+∞ }

equipped with the Luxemburg norm

‖u‖H=inf{ τ>0:ρH(uτ)≤1 }.

From Colasuonno-Squassina [8, Proposition 2.14] we know that the space L^ℌ(Ω) is a reflexive Banach space. Moreover, we need the seminormed space

Lμq(Ω)={ u∣u:Ω→ℝ is measurable and ∫Ωμ(x)|u|q dx<+∞ },

which is endowed with the seminorm

‖u‖q,μ=(∫Ωμ(x)|u|q dx)1q.

Analogously, we define Lμq(Ω;ℝN).

The Musielak-Orlicz Sobolev space W^1,ℌ(Ω) is defined by

W1,H(Ω)={ u∈LH(Ω):|∇u|∈LH(Ω) }

equipped with the norm

‖u‖1,H=‖∇u‖H+‖u‖H,

where ‖∇u‖_H = |∇u| ‖_ℌ. As before, we know that W^1,ℌ(Ω) is a reflexive Banach space.

The following proposition states the main embedding results for the spaces L^ℌ(Ω) and W^1,ℌ(Ω). We refer to Crespo-Blanco-Gasiński-Harjulehto-Winkert [11, Proposition 2.17].

Proposition 2.1

Let (2.3) be satisfied and let

(2.4) p⋆:=NpN−p and p⋆:=(N−1)pN−p

be the critical exponents to p, see (2.1) and (2.2) for r = p. Then the following embeddings hold:

L^ℌ(Ω) → L^r(Ω) and W^1,ℌ(Ω) → W^1,r(Ω) are continuous for all r ∈ [1, p];
W^1,ℌ(Ω) → L^r(Ω) is continuous for all r ∈ [1, p^*];
W^1,ℌ(Ω) → L^r(Ω) is compact for all r ∈ [1, p^*);
W^1,ℌ(Ω) → L^r(∂Ω) is continuous for all r ∈ [1, p_*];
W^1,ℌ(Ω) → L^r(∂Ω) is compact for all r ∈ [1, p_*);
LH(Ω)↔Lμq(Ω) is continuous.

We equip the space W^1,ℌ(Ω) with the equivalent norm

‖u‖0:=inf{ λ>0:∫Ω[ (|∇u|λ)p+μ(x)(|∇u|λ)q+(|u|λ)p+μ(x)(|u|λ)q ]dx≤1 }.

For u ∈ W^1,ℌ(Ω) let

(2.5) ρ^H(u)=∫Ω(|∇u|p+μ(x)|∇u|q)dx+∫Ω(|u|p+μ(x)|u|q)dx.

Based on the proof of Liu-Dai [19, Proposition 2.1] we have the following relations between the norm ·‖₀ and the modular function ˆρ^ℌ, see also Crespo-Blanco-Gasiński-Harjulehto-Winkert [11, Proposition 2.16].

Proposition 2.2

Let (2.3) be satisfied, let y ∈ W^1,_ℌ(Ω) and let ρˆ_H be defined as in (2.5).

If y ≠0, then ‖y‖₀ = λ if and only if ρ^H(yλ)=1 ;
‖y‖₀ < 1 (resp.> 1, = 1) if and only if ρˆ_ℌ(y) < 1 (resp.> 1, = 1);
If ‖y‖0<1, then ‖y‖0q⩽ρ^H(y)⩽‖y‖0p;
If ‖y‖0>1 , then ‖y‖0p⩽ρ^H(y)⩽‖y‖0q;
‖y‖0→0 if and only if ρ^H(y)→0;
‖y‖₀ → +∞if and only if ˆρ_ℌ(y) → +∞.

Let us recall some definitions which we will need in the next sections.

Definition 2.3

Let (X, · ‖_X) be a reflexive Banach space, X^* its dual space and denote by · , · its duality pairing. Let A: X → X^*, then A is called

to satisfy the (S₊)-property if u_n u in X and lim sup_n_→∞Au_n , u_n − u ≤ 0 imply u_n → u in X;
pseudomonotone if u_n u in X and lim sup_n_→∞Au_n , u_n − u ≤ 0 imply Au_n Au and Au_n , u_n → Au, u;
coercive if

lim‖u‖X→∞〈Au,u〉‖u‖X=∞.

Remark 2.4

The classical definition of pseudomonotonicity is the following one: From u_n u in X and lim sup_n_→∞Au_n , u_n − u ≤ 0 we have

liminfn→∞〈 Aun,un−v 〉≥〈Au,u−v〉 for all v∈X.

This definition is equivalent to the one in Definition 2.3(ii) when the operator is bounded. Since we are only considering bounded operators, we will use the one in Definition 2.3(ii).

The following surjectivity result for pseudomonotone operators will be used in Section 3. It can be found, for example, in Papageorgiou-Winkert [30, Theorem 6.1.57].

Theorem 2.5

Let X be a real, reflexive Banach space, let A: X → X^* be a pseudomonotone, bounded, and coercive operator, and let b ∈ X^*. Then, a solution to the equation Au = b exists.

Let A: W^1,ℌ(Ω) → W^1,ℌ(Ω)^* be the nonlinear map defined by

(2.6) ⟨ A ( u ) , φ ⟩ H = ∫ Ω | ∇ u | p − 2 ∇ u + μ ( x ) | ∇ u | q − 2 ∇ u ⋅ ∇ φ d x + ∫ Ω | u | p − 2 u + μ ( x ) | u | q − 2 u φ d x

for all u, φ ∈ W^1,ℌ(Ω), where · , · _H is the duality pairing between W^1,ℌ(Ω) and its dual space W^1,ℌ(Ω)^*. The operator A: W^1,ℌ(Ω) → W^1,ℌ(Ω)^* has the following properties, see Crespo-Blanco-Gasiński-Harjulehto-Winkert [11, Proposition 3.5].

Proposition 2.6

Let (2.3) be satisfied. Then, the operator A defined by (2.6) is bounded (that is, it maps bounded sets into bounded sets), continuous, strictly monotone (hence maximal monotone) and it is of type (S₊).

For s ∈ ℝ, we set s^± = max{±s, 0} and for u ∈ W^1,ℌ(Ω) we define u^±(·) = u(·)^±. We have

u±∈W1,H(Ω), |u|=u++u−, u=u+−u−.

For r > 1 we write r′=rr−1.

Further, we denote by C1(Ω¯)+ the positive cone

C1(Ω¯)+={ u∈C1(Ω¯):u(x)≥0 for all x∈Ω¯ }

of the ordered Banach space C1(Ω¯). This cone has a nonempty interior given by

int(C1(Ω¯)+)={ u∈C1(Ω¯):u(x)>0 for all x∈Ω¯ }.

Let us now recall some basic facts about the spectrum of the negative r-Laplacian with Robin and Steklov boundary condition, respectively, for 1 < r < ∞. We refer to the paper of Lê [18]. The r-Laplacian eigenvalue problem with Robin boundary condition is given by

(2.7) −Δru=λ|u|r−2u in Ω,|∇u|r−2∇u⋅v=−β|u|r−2u on ∂Ω,

where β > 0. We know that problem (2.7) has a smallest eigenvalue λ1,r,βR>0 which is isolated, simple and it can be variationally characterized by

(2.8) λ1,r,βR=infu∈W1,r(Ω)∣{0}∫Ω|∇u|r dx+β∫∂Ω|u|r dσ∫Ω|u|r dx.

By u1,r,βR we denote the normalized (that is, ‖ u1,r,βR ‖r=1 ) positive eigenfunction corresponding to λ1,r,βR. We know that u1,r,βR∈int(C1(Ω¯)+)

Further, we recall the r-Laplacian eigenvalue problem with Steklov boundary condition which is given by

(2.9) −Δru=−|u|r−2u in Ω,|∇u|r−2∇u⋅v=λ|u|r−2u on ∂Ω.

As before, problem (2.9) has a smallest eigenvalue λ1,rS>0 which is isolated, simple and which can be characterized by

(2.10) λ1,rS=infu∈W1,r(Ω)\{0}∫Ω|∇u|r dx+∫Ω|u|r dx∫∂Ω|u|r dσ.

The first eigenfunction associated to the first eigenvalue λ1,rS u1,rS will be denoted by and we can assume it is normalized, that is, ‖ u1,rS ‖r,∂Ω=1. We have u1,rS∈int(C1(Ω¯)+).

3 Existence results in case of convection

In this section we are interested in the existence of a solution of problem (1.1) depending on the first eigenvalues of the Robin and Steklov eigenvalue problems of the p-Laplacian. We choose

h1(x,s,ξ)=f(x,s,ξ)−|s|p−2s−μ(x)|s|q−2s for a. a. x∈Ω, h2(x,s)=g(x,s)−ζ|s|p−2s for a. a. x∈∂Ω,

for all s ∈ R and for all ξ ∈ ℝ^N with ζ > 0 specified later and Carathéodory functions f and g characterized in hypotheses (ℌ1) below. Then (1.1) becomes

(3.1) −div(|∇u|p−2∇u+μ(x)|∇u|q−2∇u)=f(x,u,∇u)−|u|p−2u−μ(x)|u|q−2u in Ω, (|∇u|p−2∇u+μ(x)|∇u|q−2∇u)⋅v=g(x,u)−ζ|u|p−2u on ∂Ω,

where we assume the following hypotheses:

(H1) The mappings f : Ω × R × ℝ^N → R and g : ∂Ω × R → R are Carathéodory functions with f (x, 0, 0) ≠ 0 for a. a. x ∈ Ω such that the following conditions are satisfied:

There exist α₁ ∈ L rr11 −1 (Ω), α₂ ∈ L rr22 −1 (∂Ω) and a₁, a₂, a₃ ≥ 0 such that

|f(x,s,ξ)|≤a1|ξ|pr1−1r1+a2|s|r1−1+α1(x) for a. a. x∈Ω, |g(x,s)|≤a3|s|r2−1+α2(x) for a. a. x∈∂Ω,

for all s ∈ R and for all ξ ∈ ℝ^N, where 1 < r₁ < p^* and 1 < r₂ < p_* with the critical exponents p^* and p_* stated in (2.4).
There exist w₁ ∈ L¹(Ω), w₂ ∈ L¹(∂Ω) and b₁, b₂, b₃ ≥ 0 such that

f(x,s,ξ)s≤b1|ξ|p+b2|s|p+w1(x) for a. a. x∈Ω, g(x,s)s≤b3|s|p+ω2(x) for a. a. x∈∂Ω,

for all s ∈ R and for all ξ ∈ ℝ^N.

A function u ∈ W^1,ℌ(Ω) is called a weak solution of problem (3.1) if

(3.2) ∫Ω(|∇u|p−2∇u+μ(x)|∇u|q−2∇u)⋅∇φdx+∫Ω(|u|p−2u+μ(x)|u|q−2u)φdx=∫Ωf(x,u,∇u)φdx+∫∂Ωg(x,u)φdσ−ζ∫∂Ω|u|p−2uφdσ

is satisfied for all φ ∈ W^1,ℌ(Ω). It is clear that this definition is well-defined.

The main result in this section is the following one.

Theorem 3.1

Let hypotheses (2.3) and (ℌ1) be satisfied. Then, there exists a nontrivial weak solution u^∈ W1,H(Ω)∩L∞(Ω) of problem (3.1) provided one of the following assertions is satisfied:

b1+b2(λ1,p,βR)−1<1 and b2β(λ1,p,βR)−1+b3<ζ;
max{ b1,b2 }+b3(λ1,pS)−1<1 and ζ≥0.

Here λ1,p,βR is the first eigenvalue of the p-Laplacian with Robin boundary condition with β>0 and λ1,pS stands for the first eigenvalue of the p-Laplacian with Steklov boundary condition, see (2.7) and (2.9), respectively.

Proof. Let N˜f:W1,H(Ω)⊂Lr1(Ω)→Lr1′(Ω) and N˜g:Lr2(∂Ω)→Lr2′(∂Ω) be the Nemytskij operators corresponding to the functions f : Ω × R × ℝ^N → R and g : ∂Ω × R → ℝ, respectively. Furthermore, we denote by i^* : L^r^′₁ (Ω) → W^1,ℌ(Ω)^* the adjoint operator of the embedding i : W^1,ℌ(Ω) → L^r₁ (Ω) and j^* : L^r^′₂ (∂Ω) → W^1,ℌ(Ω)^* stands for the adjoint operator of the embedding j : W^1,ℌ(Ω) → L^r2 (∂Ω). Then we define

Nf:=i⋆°N˜f:W1,H(Ω)→W1,H(Ω)⋆,Ng:=j⋆°N˜g°j:W1,H(Ω)→W1,H(Ω)⋆,

which are both bounded and continuous operators due to hypothesis (ℌ1)(i). Moreover, we define N_ζ : W^1,ℌ(Ω) → W^1,ℌ(Ω)^* by

Nζ:=iζ⋆°(ζ|⋅|p−2⋅)°iζ,

where iζ⋆:Lp′(Ω)→W1,H(Ω)⋆ is the adjoint operator of the embedding i_ζ : W^1,ℌ(Ω) → L^p(Ω).

Now we can define the operator A: W^1,ℌ(Ω) → W^1,ℌ(Ω)^* given by

A(u):=A(u)−Nf(u)−Ng(u)+Nζ(u).

Taking the growth conditions in (ℌ1)(i) into account, it is clear that A: W^1,ℌ(Ω) → W^1,ℌ(Ω)^* maps bounded sets into bounded sets. In order to show the pseudomonotonicity, let {u_n}_n∈N ⊂ W^1,ℌ(Ω) be such that

(3.3) un→u in W1,H(Ω) and limsupn→∞〈 Aun,un−u 〉H≤0.

From the compact embeddings W^1,ℌ(Ω) → L^ˆr(Ω) for any ˆr < p^* and W^1,ℌ(Ω) → L^˜r(∂Ω) for any ˜r < p_*, see Proposition 2.1(iii) and (v), along with (3.3) we have

un→u in Lr1(Ω) and un→u in Lr2(∂Ω),Lp(∂Ω).

Applying the growth conditions in (ℌ1)(i) along with Hölder’s inequality gives

∫Ωf(x,un,∇un)(un−u)dx≤a1∫Ω| ∇un |pr1−1r1| un−u |dx+a2∫Ω| un |r1−1| un−u |dx+∫Ω| α1(x) || un−u |dx≤a1‖ ∇un ‖ppr1−1r1‖ un−u ‖r1+a2‖ un ‖r1r1−1‖ un−u ‖r1+‖ α1 ‖r1r1−1‖ un−u ‖r1→0

and

∫∂Ωg(x,un)(un−u)dσ≤a3∫∂Ω| un |r2−1| un−u |dσ+∫∂Ω| α2(x) || un−u |dσ ≤a3‖ un ‖r2,∂Ωr2−1‖ un−u ‖r2,∂Ω+‖ α2 ‖r2r2−1,∂Ω‖ un−u ‖r2,∂Ω→0.

Furthermore, again by Hölder’s inequality, we have

ζ∫∂Ω| un |p−2un(un−u)dσ≤ζ‖ un ‖p,∂Ωp−1‖ un−u ‖p,∂Ω→0.

Replacing u by u_n and φ by u_n − u in the weak formulation in (3.2) and using the considerations above leads to

(3.4) limsupn→∞〈 A(un),un−u 〉H=limsupn→∞〈 A(un),un−u 〉H≤0.

From Proposition 2.6 we know that A fulfills the (S₊)-property. Therefore, from (3.3) and (3.4) we conclude that

un→u in W1,H(Ω).

Since A is continuous we have A(u_n)→ A(u) in W^1,ℌ(Ω)^* which shows that A is pseudomonotone. Let us now prove that A: W^1,ℌ(Ω) → W^1,ℌ(Ω)^* is coercive. We distinguish between two cases.

Case I: Condition (A) is satisfied.

From the p-Laplace eigenvalue problem with Robin boundary condition, see (2.7) and (2.8) for r = p, we know that

(3.5) ‖u‖pp≤(λ1,p,βR)−1(‖∇u‖pp+β‖u‖p,∂Ωp) for all u∈W1,p(Ω).

Let u ∈ W^1,ℌ(Ω) be such that ‖u‖⁰ > 1 and note that W₁^,ℌ(Ω) ⊆ W^1,p(Ω). Then, from (ℌ1)(ii), (3.5), (A) and Proposition 2.2(iv) we obtain

〈 A ( u ) , u 〉 H = ∫ Ω | ∇ u | p − 2 ∇ u + μ ( x ) | ∇ u | q − 2 ∇ u ⋅ ∇ u d x + ∫ Ω | u | p − 2 u + μ ( x ) | u | q − 2 u u d x − ∫ Ω f ( x , u , ∇ u ) u d x − ∫ ∂ Ω g ( x , u ) u d σ + ζ ∫ ∂ Ω u p d σ ≥ ∥ ∇ u ∥ p p + ∥ ∇ u ∥ q , μ q + ∥ u ∥ p p + ∥ u ∥ q , μ q − b 1 ∥ ∇ u ∥ p p − b 2 ∥ u ∥ p p − ω 1 1 − b 3 ∥ u ∥ p , ∂ Ω p − ω 2 1 , ∂ Ω + ζ ∥ u ∥ p , ∂ Ω p ≥ 1 − b 1 − b 2 λ 1 , p , β R − 1 ∥ ∇ u ∥ p p + ∥ u ∥ p p + ∥ ∇ u ∥ q , μ q + ∥ u ∥ q , μ q + ζ − b 2 β λ 1 , p , β R − 1 − b 3 ∥ u ∥ p , ∂ Ω p − ω 1 1 − ω 2 1 , ∂ Ω ≥ 1 − b 1 − b 2 λ 1 , p , β R − 1 ∥ ∇ u ∥ p p + ∥ u ∥ p p + ∥ ∇ u ∥ q , μ q + ∥ u ∥ q , μ q − ω 1 1 − ω 2 1 , ∂ Ω = 1 − b 1 − b 2 λ 1 , p , β R − 1 ρ ^ H ( u ) − ω 1 1 − ω 2 1 , ∂ Ω ≥ 1 − b 1 − b 2 λ 1 , p , β R − 1 ∥ u ∥ 0 p − ω 1 1 − ω 2 1 , ∂ Ω

This shows the coercivity of A.

Case II: Condition (B) is satisfied.

From the Steklov p-Laplace eigenvalue problem, see (2.9) and (2.10) for r = p, we have the inequality

(3.6) ‖u‖p,∂Ωp≤(λ1,pS)−1(‖∇u‖pp+‖u‖pp) for all u∈W1,p(Ω).

As before, let u ∈ W^1,ℌ(Ω) be such that ‖u‖₀ > 1 and note again that W^1,ℌ(Ω) ⊆ W^1,p(Ω). Applying (ℌ1)(ii), (3.6), (B) and Proposition 2.2(iv) one gets

〈A(u),u〉H=∫Ω(|∇u|p−2∇u+μ(x)|∇u|q−2∇u)⋅∇u dx+∫Ω(|u|p−2u+μ(x)|u|q−2u)u dx

−∫Ωf(x,u,∇u)u dx−∫∂Ωg(x,u)u dσ+ζ∫∂Ω| u |p dσ≥‖∇u‖pp+‖∇u‖q,μq+‖u‖pp+‖u‖q,μq−b1‖∇u‖pp−b2‖u‖pp−‖ ω1 ‖1 −b3‖u‖p,∂Ωp−‖ ω2 ‖1,∂Ω+ζ‖u‖p,∂Ωp≥(1−max{ b1,b2 }−b3(λ1,pS)−1)(‖∇u‖pp+‖u‖pp)+‖∇u‖q,μq+‖u‖q,μq−‖ ω1 ‖1−‖ ω2 ‖1,∂Ω≥(1−max{ b1,b2 }−b3(λ1,pS)−1)ρ^H(u)−‖ ω1 ‖1−‖ ω2 ‖1,∂Ω≥(1−max{ b1,b2 }−b3(λ1,pS)−1)‖u‖0p−‖ ω1 ‖1−‖ ω2 ‖1,∂Ω.

Hence, A: W^1,ℌ(Ω) → W^1,ℌ(Ω)^* is again coercive.

We have shown that A: W^1,ℌ(Ω) → W^1,ℌ(Ω)^* is a bounded, pseudomonotone and coercive operator. From Theorem 2.5 we find an element ˆu ∈ W^1,ℌ(Ω) such that A(ˆu) = 0 with ˆu ≠ 0 since f (x, 0, 0) ≠ 0 for a. a. x ∈ Ω. In view of the definition of A, we see that ˆu turns out to be a nontrivial weak solution of problem (3.1). Similar to Theorem 3.1 of Gasiński-Winkert [17] we can show the boundedness of ˆu. The proof is complete.

4 Constant sign solutions for superlinear perturbations

In this section we are interested in constant sign solutions for problems of type (1.1) without convection term but with superlinear nonlinearities. We are going to consider the cases of the dependence on Robin and Steklov eigenvalues separately. We start with the Steklov case and set

h1(x,s,ξ)=−ϑ|s|p−2s−μ(x)|s|q−2s−f(x,s) for a. a. x∈Ω, h2(x,s)=ζ|s|p−2s−g(x,s) for a.a. x∈∂Ω,

for all s ∈ ℝ, ϑ, ζ > 0 to be specified and Carathéodory functions f and g which satisfy hypotheses (ℌ2) below.

With this choice, (1.1) can be written as

(4.1) −div(|∇u|p−2∇u+μ(x)|∇u|q−2∇u)=−θ|u|p−2u−μ(x)|u|q−2u−f(x,u) in Ω, (|∇u|p−2∇u+μ(x)|∇u|q−2∇u)⋅v=ζ|u|p−2u−g(x,u) on ∂Ω,

where the following conditions are supposed:

(H2) The nonlinearities f : Ω × R → R and g : ∂Ω × R → R are assumed to be Carathéodory functions which satisfy the subsequent hypotheses:

f and g are bounded on bounded sets.
It holds

lims→±∞f(x,s)|s|q−2s=+∞ uniformly for a. a. x∈Ω,lims→±∞g(x,s)|s|q−2s=+∞ uniformly for a. a. x∈∂Ω.
It holds

lims→0f(x,s)|s|q−2s=0 uniformly for a. a. x∈Ω,lims→0g(x,s)|s|p−2s=0 uniformly for a.a. x∈∂Ω.

We say that u ∈ W^1,ℌ(Ω) is a weak solution of problem (4.1) if

∫Ω(|∇u|p−2∇u+μ(x)|∇u|q−2∇u)⋅∇φdx+∫Ω(ϑ|u|p−2u+μ(x)|u|q−2u)φdx=∫Ω(−f(x,u))φdx+∫∂Ω(ζ|u|p−2u−g(x,u))φdσ

is fulfilled for all φ ∈ W^1,ℌ(Ω).

The following theorem states the existence of constant sign solutions where the parameter ζ depends on the first Steklov eigenvalue for the p-Laplacian, namely λ1,pS.

Theorem 4.1

Let hypotheses (2.3) and (ℌ2) be satisfied. Furthermore, let ϑ ∈ (0, 1] and let ζ>λ1,pS with λ1,pS being the first eigenvalue of the Steklov eigenvalue problem of the p-Laplacian stated in (2.9). Then, problem (4.1) has at least two nontrivial weak solutions u₀, v₀ ∈ W^1,ℌ(Ω) ∩ L^∞(Ω) such that u₀ ≥ 0 and v₀ ≤ 0.

Proof. From hypothesis (ℌ2)(ii) we know that we can find constants M ₁, M₂ = M ₂(ζ ) > 1 such that

(4.2) f(x,s)s≥|s|q for a.a. x∈Ω and all |s|≥M1,g(x,s)s≥ζ|s|q for a.a. x∈Ω and all |s|≥M2.

We set M₃ = max (M₁, M₂) and take a constant function u ≡ ς ∈ [M₃, +∞). Applying (4.2), p < q and M₃ > 1 yields

(4.3) 0≥−f(x,u¯) for a. a. x∈Ω and 0≥ζu¯p−1−g(x,u¯) for a.a. x∈∂Ω.

Analogously, we can choose v ≡ −ς in order to get

0≤−f(x,v_) for a.a. x∈Ω and 0≤ζ|v_|p−2v_−g(x,v_) for a.a. x∈∂Ω.

Now, we introduce the cut-off functions θ±:Ω×ℝ→ℝ and θζ±:∂Ω×ℝ→ℝ defined by

(4.4) θ+(x,s)={ 0 if s<0−f(x,s) if 0≤s≤u¯,−f(x,u¯) if u¯<s θζ+(x,s)={ 0 if s<0ζsp−1−g(x,s) if 0≤s≤u¯ζu¯p−1−g(x,u¯) if u¯<s ,θ−(x,s)={ −f(x,v_) if s<v_−f(x,s) if v_≤s≤00 if 0<s ,θζ−(x,s)={ ζ|∣_|p−2v_−g(x,v_) if s<v_ζ|s|p−2s−g(x,s) if v_≤s≤00 if 0<s ,

which are Carathéodory functions. We set

Θ±(x,s)=∫0sθ±(x,t)dt and Θζ±(x,s)=∫0sθζ±(x,t)dt

Now we consider the C ¹-functionals Γ^± : W^1,ℌ(Ω) → R defined by

Γ±(u)=1p‖∇u‖pp+1q‖∇u‖q,μq+ϑp‖u‖pp+1q‖u‖q,μq−∫ΩΘ±(x,u)dx−∫∂ΩΘζ±(x,u)dσ.

Furthermore, we write F(x,s)=∫0sf(x,t)dt and G(x,s)=∫0sg(x,t)dt.

We first investigate the existence of the nonnegative solution. Due to the truncations in (4.4) it is clear that the functional Γ⁺ is coercive and also sequentially weakly lower semicontinuous. Hence, its global minimizer u₀ ∈ W^1,ℌ(Ω) exists, that is

Γ+(u0)=inf[ Γ+(u):u∈W1,H(Ω) ].

From hypotheses (ℌ2)(iii), for given ε₁, ε₂ > 0, there exist δ₁ = δ₁(ε₁), δ₂ = δ₂(ε₂) ∈ (0, u) such that

(4.5) F(x,s)≤ε1q|s|q for a.a. x∈Ω and for all |s|≤δ1,G(x,s)≤ε2p|s|p for a.a. x∈∂Ω and for all |s|≤δ2.

We set δ := min(δ₁, δ₂). Recall that u1,pS is the first eigenfunction corresponding to the first eigenvalue λ1,pS of the eigenvalue problem of the p-Laplacian with Steklov boundary condition, see (2.9). We may suppose that it is normalized, that is, ‖ u1,pS ‖p,∂Ω=1. Since u1,pS∈int(C1(Ω¯)+), we may choose t ∈ (0, 1) small enough such that tu1,pS(x)∈[0,δ] for all x ∈ Ω. Because of (4.4), (4.5) and δ < u it follows that

(4.6) Γ + t u 1 , p S = 1 p ∇ t u 1 , p S p p + 1 q ∇ t u 1 , p S q , μ q + ϑ p t u 1 , p S p p + 1 q t u 1 , p S q , μ q − ∫ Ω Θ + x , t u 1 , p S d x − ∫ ∂ Ω Θ ζ + x , t u 1 , p S d σ ≤ t p p λ 1 , p S + t q q ∇ u 1 , p S q , μ q + t q q u 1 , p S q , μ q + ∫ ∂ F x , t u 1 , p S d x − ζ t p p + ∫ ∂ Ω G x , t u 1 , p S d σ ≤ t p p λ 1 , p S + t q q ∇ u 1 , p S q , μ q + t q q u 1 , p S q , μ q + ε 1 t q q u 1 , p S q q − ζ t p p + ε 2 t p p = t p λ 1 , p S − ζ + ε 2 p + t q ∇ u 1 , p S q , μ q + u 1 , p S q , μ q + ε 1 u 1 , p S q 4 q .

By assumption, we know that ζ>λ1,pS. So we may choose ε₁, ε₂ > 0 such that

0<ε1<∞ and 0<ε2<ζ−λ1,pS.

From this choice and since p < q we obtain from (4.6)

Γ+(tu1,pS)<0 for all sufficiently small t>0.

Therefore, we know now that

Γ+(u0)<0=Γ+(0).

Hence, u₀ ≠0.

Since u₀ is a global minimizer of Γ⁺ we have (Γ⁺)^′(u₀) = 0, that is,

(4.7) ∫Ω(| ∇u0 |p−2∇u0+μ(x)| ∇u0 |q−2∇u0)⋅∇φdx +∫Ω(ϑ| u0 |p−2u0+μ(x)| u0 |q−2u0) φdx=∫Ωθ+(x,u0)φdx+∫∂Ωθζ+(x,u0)φdσ

for all φ ∈ W^1,ℌ(Ω). First we take φ=−u0−∈W1,H(Ω) as test function in (4.7). We obtain

‖ ∇u0− ‖pp+‖ ∇u0− ‖q,μq+‖ u0− ‖pp+‖ u0− ‖q,μq=0,

which yields u0−=0 and so u₀ ≥ 0. Second we choose φ = (u₀ − u)⁺ ∈ W^1,ℌ(Ω) as test function in (4.7) which results in

(4.8) ∫Ω(| ∇u0 |p−2∇u0+μ(x)| ∇u0 |q−2∇u0)⋅∇(u0−u¯)+dx +∫Ω(ϑu0p−1+μ(x)u0q−1)(u0−u¯)+dx=∫Ωθ+(x,u0)(u0−u¯)+dx+∫∂Ωθζ+(x,u0)(u0−u¯)+dσ=∫Ω(−f(x,u¯))(u0−u¯)+dx+∫∂Ω(ζu¯p−1−g(x,u¯))(u0−u¯)+dσ≤0,

by (4.3). First note that

(4.9) ∫Ω(| ∇u0 |p−2∇u0+μ(x)| ∇u0 |q−2∇u0)⋅∇(u0−u¯)+dx≥ϑ∫Ω(| ∇(u0−u¯)+ |p+μ(x)| ∇(u0−u¯)+ |q)dx

Since u0>u¯>1 on the set { u0>u¯ } we have

(4.10) ∫Ω(ϑu0p−1+μ(x)u0q−1)(u0−u¯)+dx≥ϑ∫{ u0>u¯ }(u0p−1+μ(x)u0q−1)(u0−u¯)dx≥ϑ∫{ u0>u¯ }((u0−u¯)p−1+μ(x)(u0−u¯)q−1)(u0−u¯)dx=ϑ∫Ω(((u0−u¯)+)p+μ(x)((u0−u¯)+)q)dx.

Combining (4.8) with (4.9) as well as (4.10) and using Proposition 2.2(iii), (iv) implies that

ϑmin{ ‖ (u0−u¯)+ ‖0p,‖ (u0−u¯)+ ‖0q }≤ϑρ^H((u0−u¯)+)≤0.

Hence, u₀ ≤ u and so u₀ ∈ [0, u]. By the definition of the truncations in (4.4)we see that u₀ ∈ W^1,ℌ(Ω)∩L^∞(Ω) turns out to be a weak solution of our original problem (4.1).

For the nonpositive solution we consider the functional Γ⁻ : W^1,ℌ(Ω) → R and show in the same way that it has a global minimizer v₀ ∈ W^1,ℌ(Ω) which belongs to [v_, 0].

Let us study now the case when the solutions depend on the first Robin eigenvalue. We set

h1(x,s,ξ)=(ζ−ϑ)|s|p−2s−μ(x)|s|q−2s−f(x,s) for a.a. x∈Ω, h2(x,s)=−β|s|p−2s for a.a. x∈∂Ω,

for all s ∈ R with parameters ζ > ϑ > 0 to be specified, β > 0 is the same parameter as in the Robin eigenvalue problem and f is a Carathéodory function. Then, problem (1.1) becomes

(4.11) −div(|∇u|p−2∇u+μ(x)|∇u|q−2∇u)=(ζ−ϑ)|u|p−2u−μ(x)|u|q−2u−f(x,u) in Ω, (|∇u|p−2∇u+μ(x)|∇u|q−2∇u)⋅v=−β|u|p−2u on ∂Ω,

where f satisfies the following assumptions:

(H3) The function f : Ω × R → R is a Carathéodory function such that:

f is bounded on bounded sets.
It holds

lims→±∞f(x,s)|s|q−2s=+∞ uniformly for a. a. x∈Ω.
It holds

lims→0f(x,s)|s|p−2s=0 uniformly for a. a. x∈Ω.

We have the following multiplicity result concerning problem (4.11).

Theorem 4.2

Let hypotheses (2.3) and (H3) be satisfied. Further, let ζ>λ1,p,βR+ϑ with ϑ > 0 and λ1,p,βR being the first eigenvalue of the Robin eigenvalue problem of the p-Laplacian with β > 0stated in (2.7). Then, problem (4.11) has at least two nontrivial weak solutions u₁, v₁ ∈ W^1,ℌ(Ω) ∩ L^∞(Ω) such that u₁ ≥ 0 and v₁ ≤ 0.

Proof. Taking hypothesis (H3)(ii) into account we find a constant M = M (ζ ) > 1 such that

(4.12) f(x,s)s≥ζ|s|q for a.a. x∈Ω and all |s|≥M.

As in the proof of Theorem 4.1, by (4.12), we can take constant functions u¯∈(M,+∞) and v_≡−u¯ such that

(4.13) 0≥ζu¯p−1−f(x,u¯) for a. a. x∈Ω and 0≤ζ|v_|p−2v_−f(x,v_) for a.a. x∈Ω,

because p < q and M > 1.

Then we define truncations ψζ±:Ω×ℝ→ℝ and ψβ±:∂Ω×ℝ→ℝ as follows

(4.14) ψζ+(x,s)={ 0 if s<0ζsp−1−f(x,s) if 0≤s≤u¯ζu¯p−1−f(x,u¯) if u¯<s ,ψβ+(x,s)={ 0 if s<0−βsp−1 if 0≤s≤u¯−βu¯p−1 if u¯<s ,ψζ−(x,s)={ ζ|v_|p−2v_−f(x,v_) if s<v_ζ|s|p−2s−f(x,s) if v_≤s≤00 if 0<s ,ψβ−(x,s)={ −β|v_|p−2v_ if s<v_−β|s|p−2s if v_≤s≤00 if 0<s .

We set

Ψζ±(x,s)=∫0sψζ±(x,t)dt and Ψβ±(x,s)=∫0sψβ±(x,t)dt

and introduce the C ¹-functionals Π^± : W^1,H(Ω) → R given by

Π±(u)=1p‖∇u‖pp+1q‖∇u‖q,μq+ϑp‖u‖pp+1q‖u‖q,μq−∫ΩΨζ±(x,u)dx−∫∂ΩΨβ±(x,u)dσ

As before, we define F(x,s)=∫0sf(x,t)dt .

We start with the existence of a nonnegative solution. Because of (4.14) we know that the functional Π⁺ is coercive and also sequentially weakly lower semicontinuous. Therefore, we find an element u₁ ∈ W^1,H(Ω) such that

Π+(u1)=inf[ Π+(u):u∈W1,H(Ω) ].

By hypothesis (H3)(iii), we find for every ε > 0 a number δ ∈ (0, u) such that

(4.15) F(x,s)≤εp|s|p for a.a. x∈Ω and for all |s|≤δ.

We recall that u1,p,βR is the first eigenfunction corresponding to the first eigenvalue λ1,p,βR of the eigenvalue problem of the p-Laplacian with Robin boundary condition, see (2.7). Without any loss of generality we can assume that u1,p,βR is normalized (that is, ‖ u1,p,βR ‖p=1 ) and because of u1,p,βR∈int(C1(Ω¯)+) we choose t ∈ (0, 1) sufficiently small such that tu1,p,βR(x)∈[0,δ] for all x∈Ω¯. Applying (4.14), (4.15), δ < u and ϑ > 0 gives

(4.16) Π + t u 1 , p , β R = 1 p ∇ t u 1 , p , β R p p + 1 q ∇ t u 1 , p , β R q , μ q + ϑ p t u 1 , p , β R p p + 1 q t u 1 , p , β R q , μ q − ∫ Ω Ψ ζ + x , t u 1 , p , β R d x − ∫ ∂ Ω Ψ β + x , t u 1 , p , β R d σ ≤ t p p λ 1 , p , β R − β t p p u 1 , p , β R p , ∂ Ω p + t q q ∇ u 1 , p , β R q , μ q + t p ϑ p + t q q u 1 , p , β R q , μ q − ζ t p p + ∫ Ω F x , t u 1 , p , β R d x + β t p p u 1 , p , β R p , ∂ Ω p ≤ t p p λ 1 , p , β R + t q q ∇ u 1 , p , β R q , μ q + t p ϑ p + t q q u 1 , p , β R q , μ q − ζ t p p + ε t p p ≤ t p λ 1 , p , β R + ϑ − ζ + ε p + t q ∇ u 1 , p , β R q , μ q + u 1 , p , β R q , μ q .

Due to ζ>λ1,p,βR+ϑ and p<q one has from (4.16) for ε∈(0,ζ−λ1,p,βR−ϑ) that

Π+(tu1,p,βR)<0 for all sufficiently small t>0.

Hence, Π⁺ (u₁)< 0 = Π⁺ (0) and so u¹ ≠0.

We have (Π⁺)^′(u₁) = 0, that is,

(4.17) ∫Ω(| ∇u1 |p−2∇u1+μ(x)| ∇u1 |q−2∇u1)⋅∇φdx +∫Ω(θ| u1 |p−2u1+μ(x)| u1 |q−2u1)φdx=∫Ωψζ+(x,u1)φdx+∫∂Ωψβ+(x,u1)φdσ

for all φ _∈ W^1,ℌ(Ω). As done in the proof of Theorem 4.1 we take φ = −u₁∈ W^1,ℌ(Ω) and φ = (u₁ − u)⁺ ∈ W^1,ℌ(Ω) as test functions in (4.17) which gives us 0 ≤ u₁ ≤ u, see (4.13). Hence, by the definition of the truncations in (4.14) we see that u₁ ∈ W^1,ℌ(Ω) ∩ L^∞(Ω) solves problem (4.11).

In the same way we can show the existence of a nontrivial nonpositive solution v₁ ∈ W^1,ℌ(Ω) ∩ L^∞(Ω) by treating the functional Π⁻ : W^1,ℌ(Ω) → R instead of Π⁺ : W^1,ℌ(Ω) → ℝ.

Remark 4.3

In this section we decided to consider two different problems since in the proof of Theorem 4.1 the use of the first Robin eigenfunction would have provided a condition of the form

(4.18) λ1,p,βR+ϑ<(β+ζ)‖ u1,p,βR ‖p,∂Ωp,

which depends also on the boundary norm of the eigenfunction u1,p,βR. So the statement of Theorem 4.1 still holds true whenwe replace the assumption ζ > λ^S_1,p by (4.18) where u_ℝ_1,p,β is the first normalized (that is, ‖ u1,p,βR ‖p=1 ) eigenfunction associated to the first eigenvalue λ1,p,βR of the Robin eigenvalue problem.

samanouni@imamu.edu.sa

Acknowledgments

The authors wish to thank knowledgeable referees for their corrections and remarks. Furthermore, the authors wish to thank Ángel Crespo-Blanco for valuable comments and improvements. We acknowledge support by the German Research Foundation and the Open Access Publication Fund of TU Berlin.

Conflict of Interest: The authors declare that they have no conflict of interest.

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Received: 2021-02-18

Accepted: 2021-05-20

Published Online: 2021-07-29

This work is licensed under the Creative Commons Attribution 4.0 International License.

Existence results for double phase problems depending on Robin and Steklov eigenvalues for the p-Laplacian

Abstract

1 Introduction

2 Preliminaries

Proposition 2.1

Proposition 2.2

Definition 2.3

Remark 2.4

Theorem 2.5

Proposition 2.6

3 Existence results in case of convection

Theorem 3.1

4 Constant sign solutions for superlinear perturbations

Theorem 4.1

Theorem 4.2

Remark 4.3

Acknowledgments

References

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