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

Said El Manouni, Greta Marino and Patrick Winkert

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.

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

1 Introduction

Let Ω ⊂ ℝN, 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|p2u+μ(x)|u|q2u)=h1(x,u,u) in Ω,(|u|p2u+μ(x)|u|q2u)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 × ℝN → 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|p2u+μ(x)|u|q2u) for uW1,H(Ω)

with an appropriate Musielak-Orlicz Sobolev space W1,ℌ(Ω), see its definition in Section 2. Note that when infΩ μ > 0 or μ ≡ 0 then the operator becomes the weighted (q, p)-Laplacian or the p-Laplacian, respectively.

The energy functional J : W1,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 {xΩ : 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|p2u) )Δqu+ξ(z)|u|p2u=λf(z,u(z)) in Ω,unθ+β|u|p2u=0onΩ,

where 1 < q < p < N, ξL(Ω) is a positive potential, a(z) > 0 for a. a. zΩ and

unθ=[ a(z)|u|p2+|u|q2 ]un

with n(·) being the outward unit normal on ∂Ω. Under different assumptions it is shown that problem (1.4) admits two nontrivial solutions uλ , ˆuλW1,ℌ(Ω) for small λ > 0 such that ‖uλ1,H → +∞and ‖ˆuλ1,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|p2u) )Δqu+ξ(z)|u|p2u=f(z,u(z)) in Ω,unθ+β|u|p2u=0 on Ω,

where 1 < q < pN 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|p2u+μ(x)|u|q2u)=f(x,u)|u|p2uμ(x)|u|q2u in Ω,(|u|p2u+μ(x)|u|q2u)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))+up1+μ(x)uq1=up*1+λ(uγ1+g1(x,u)) in Ω,A(u)v=up*1+g2(x,u)on Ω,u>0 in Ω,

where

div(A(u)):=div(Fp1(u)F(u)+μ(x)Fq1(u)F(u))

is the so-called Finsler double phase operator and (ℝN , 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 W1,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 Lr(Ω) and Lr(Ω;ℝN) the usual Lebesgue spaces equipped with the norm · ‖r and for 1 < r < ∞we consider the corresponding Sobolev space W1,r(Ω) endowed with the norm ‖·‖1,r. It is known that W1,r(Ω) → Lˆr(Ω) is compact for ˆr < r* and continuous for ˆr = r*,where r* is the critical exponent of r defined by

(2.1) r={ NrNr if r<N, any l(r,) if rN.

On the boundary ∂Ω of Ω we consider the (N −1)-dimensional Hausdorff (surface) measure σ and denote by Lr(∂Ω) the boundary Lebesgue space with norm ‖·‖r,∂Ω. Fromthe definition of the tracemappingwe know that W1,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*={ (N1)rNr if r<N, any l(r,) if rN.

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 0μ()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+q1<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(Ω) is defined by

LH(Ω)={ uu:Ω is measurable and ρH(u)<+ }

equipped with the Luxemburg norm

uH=inf{ τ>0:ρH(uτ)1 }.

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

Lμq(Ω)={ uu:Ω is measurable and Ωμ(x)|u|qdx<+ },

which is endowed with the seminorm

uq,μ=(Ωμ(x)|u|qdx)1q.

Analogously, we define Lμq(Ω;N).

The Musielak-Orlicz Sobolev space W1,ℌ(Ω) is defined by

W1,H(Ω)={ uLH(Ω):|u|LH(Ω) }

equipped with the norm

u1,H=uH+uH,

where ‖∇uH = |∇u| ‖. As before, we know that W1,ℌ(Ω) is a reflexive Banach space.

The following proposition states the main embedding results for the spaces L(Ω) and W1,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:=NpNpandp:=(N1)pNp

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

1. L(Ω) → Lr(Ω) and W1,ℌ(Ω) → W1,r(Ω) are continuous for all r ∈ [1, p];

2. W1,ℌ(Ω) → Lr(Ω) is continuous for all r ∈ [1, p*];

3. W1,ℌ(Ω) → Lr(Ω) is compact for all r ∈ [1, p*);

4. W1,ℌ(Ω) → Lr(∂Ω) is continuous for all r ∈ [1, p*];

5. W1,ℌ(Ω) → Lr(∂Ω) is compact for all r ∈ [1, p*);

6. LH(Ω)Lμq(Ω) is continuous.

We equip the space W1,ℌ(Ω) with the equivalent norm

u0:=inf{ λ>0:Ω[ (|u|λ)p+μ(x)(|u|λ)q+(|u|λ)p+μ(x)(|u|λ)q ]dx1 }.

For uW1,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, see also Crespo-Blanco-Gasiński-Harjulehto-Winkert [11, Proposition 2.16].

Proposition 2.2

Let (2.3) be satisfied, let yW1,(Ω) and let ρˆH be defined as in (2.5).

1. If y ≠0, theny0 = λ if and only if ρ^H(yλ)=1 ;

2. y0 < 1 (resp.> 1, = 1) if and only if ρˆ(y) < 1 (resp.> 1, = 1);

3.  If y0<1, then y0qρ^H(y)y0p;

4.  If y0>1 , theny0pρ^H(y)y0q;

5. y00if and only if ρ^H(y)0;

6. y0 → +∞if and only if ˆρ(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: XX*, then A is called

1. to satisfy the (S+)-property if un u in X and lim supn→∞Aun , unu ≤ 0 imply unu in X;

2. pseudomonotone if un u in X and lim supn→∞Aun , unu ≤ 0 imply Aun Au and Aun , unAu, u;

3. coercive if

limuXAu,uuX=.

Remark 2.4

The classical definition of pseudomonotonicity is the following one: From un u in X and lim supn→∞Aun , unu ≤ 0 we have

liminfn Aun,unv Au,uvfor allvX.

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: XX* be a pseudomonotone, bounded, and coercive operator, and let bX*. Then, a solution to the equation Au = b exists.

Let A: W1,ℌ(Ω) → W1,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, φW1,ℌ(Ω), where · , · H is the duality pairing between W1,ℌ(Ω) and its dual space W1,ℌ(Ω)*. The operator A: W1,ℌ(Ω) → W1,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 uW1,ℌ(Ω) we define u±(·) = u(·)±. We have

u±W1,H(Ω),|u|=u++u,u=u+u.

For r > 1 we write r=rr1.

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

C1(Ω¯)+={ uC1(Ω¯):u(x)0 for all xΩ¯ }

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

int(C1(Ω¯)+)={ uC1(Ω¯):u(x)>0 for all xΩ¯ }.

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|r2u in Ω,|u|r2uv=β|u|r2u 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=infuW1,r(Ω){0}Ω|u|rdx+βΩ|u|rdσΩ|u|rdx.

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,βRint(C1(Ω¯)+)

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

(2.9) Δru=|u|r2u in Ω,|u|r2uv=λ|u|r2u 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=infuW1,r(Ω)\{0}Ω|u|rdx+Ω|u|rdxΩ|u|rdσ.

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,rSint(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|p2sμ(x)|s|q2s for a. a. xΩ,h2(x,s)=g(x,s)ζ|s|p2sfor a. a. xΩ,

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

(3.1) div(|u|p2u+μ(x)|u|q2u)=f(x,u,u)|u|p2uμ(x)|u|q2u in Ω,(|u|p2u+μ(x)|u|q2u)v=g(x,u)ζ|u|p2uon Ω,

where we assume the following hypotheses:

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

1. There exist α1L rr11 −1 (Ω), α2L rr22 −1 (∂Ω) and a1, a2, a3 ≥ 0 such that

|f(x,s,ξ)|a1|ξ|pr11r1+a2|s|r11+α1(x) for a. a. xΩ,|g(x,s)|a3|s|r21+α2(x) for a. a. xΩ,

for all s ∈ R and for all ξ ∈ ℝN, where 1 < r1 < p* and 1 < r2 < p* with the critical exponents p* and p* stated in (2.4).

2. There exist w1L1(Ω), w2L1(∂Ω) and b1, b2, b3 ≥ 0 such that

f(x,s,ξ)sb1|ξ|p+b2|s|p+w1(x) for a. a. xΩ,g(x,s)sb3|s|p+ω2(x)for a. a. xΩ,

for all s ∈ R and for all ξ ∈ ℝN.

A function uW1,ℌ(Ω) is called a weak solution of problem (3.1) if

(3.2) Ω(|u|p2u+μ(x)|u|q2u)φdx+Ω(|u|p2u+μ(x)|u|q2u)φdx=Ωf(x,u,u)φdx+Ωg(x,u)φdσζΩ|u|p2uφdσ

is satisfied for all φW1,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:

1. b1+b2(λ1,p,βR)1<1andb2β(λ1,p,βR)1+b3<ζ;

2. max{ b1,b2 }+b3(λ1,pS)1<1andζ0.

Here λ1,p,βR is the first eigenvalue of the p-Laplacian with Robin boundary condition with β>0andλ1,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 × ℝN → R and g : ∂Ω × R → ℝ, respectively. Furthermore, we denote by i* : Lr1 (Ω) → W1,ℌ(Ω)* the adjoint operator of the embedding i : W1,ℌ(Ω) → Lr1 (Ω) and j* : Lr2 (∂Ω) → W1,ℌ(Ω)* stands for the adjoint operator of the embedding j : W1,ℌ(Ω) → Lr2 (∂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 Nζ : W1,ℌ(Ω) → W1,ℌ(Ω)* by

Nζ:=iζ°(ζ||p2)°iζ,

where iζ:Lp(Ω)W1,H(Ω) is the adjoint operator of the embedding iζ : W1,ℌ(Ω) → Lp(Ω).

Now we can define the operator A: W1,ℌ(Ω) → W1,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: W1,ℌ(Ω) → W1,ℌ(Ω)* maps bounded sets into bounded sets. In order to show the pseudomonotonicity, let {un}nN ⊂ W1,ℌ(Ω) be such that

(3.3) unu in W1,H(Ω) and limsupn Aun,unu H0.

From the compact embeddings W1,ℌ(Ω) → Lˆr(Ω) for any ˆr < p* and W1,ℌ(Ω) → L˜r(∂Ω) for any ˜r < p*, see Proposition 2.1(iii) and (v), along with (3.3) we have

unu in Lr1(Ω) and unu in Lr2(Ω),Lp(Ω).

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

Ωf(x,un,un)(unu)dxa1Ω| un |pr11r1| unu |dx+a2Ω| un |r11| unu |dx+Ω| α1(x) || unu |dxa1 un ppr11r1 unu r1+a2 un r1r11 unu r1+ α1 r1r11 unu r10

and

Ωg(x,un)(unu)dσa3Ω| un |r21| unu |dσ+Ω| α2(x) || unu |dσa3 un r2,Ωr21 unu r2,Ω+ α2 r2r21,Ω unu r2,Ω0.

Furthermore, again by Hölder’s inequality, we have

ζΩ| un |p2un(unu)dσζ un p,Ωp1 unu p,Ω0.

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

(3.4) limsupn A(un),unu H=limsupn A(un),unu H0.

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

unu in W1,H(Ω).

Since A is continuous we have A(un)→ A(u) in W1,ℌ(Ω)* which shows that A is pseudomonotone. Let us now prove that A: W1,ℌ(Ω) → W1,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) upp(λ1,p,βR)1(upp+βup,Ωp) for all uW1,p(Ω).

Let uW1,ℌ(Ω) be such that ‖u0 > 1 and note that W1,ℌ(Ω) ⊆ W1,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) up,Ωp(λ1,pS)1(upp+upp) for all uW1,p(Ω).

As before, let uW1,ℌ(Ω) be such that ‖u0 > 1 and note again that W1,ℌ(Ω) ⊆ W1,p(Ω). Applying (ℌ1)(ii), (3.6), (B) and Proposition 2.2(iv) one gets

A(u),uH=Ω(|u|p2u+μ(x)|u|q2u)udx+Ω(|u|p2u+μ(x)|u|q2u)udx
Ωf(x,u,u)udxΩg(x,u)udσ+ζΩ| u |pdσupp+uq,μq+upp+uq,μqb1uppb2upp ω1 1b3up,Ωp ω2 1,Ω+ζup,Ωp(1max{ b1,b2 }b3(λ1,pS)1)(upp+upp)+uq,μq+uq,μq ω1 1 ω2 1,Ω(1max{ b1,b2 }b3(λ1,pS)1)ρ^H(u) ω1 1 ω2 1,Ω(1max{ b1,b2 }b3(λ1,pS)1)u0p ω1 1 ω2 1,Ω.

Hence, A: W1,ℌ(Ω) → W1,ℌ(Ω)* is again coercive.

We have shown that A: W1,ℌ(Ω) → W1,ℌ(Ω)* is a bounded, pseudomonotone and coercive operator. From Theorem 2.5 we find an element ˆuW1,ℌ(Ω) such that A(ˆu) = 0 with ˆu ≠ 0 since f (x, 0, 0) ≠ 0 for a. a. xΩ. 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|p2sμ(x)|s|q2sf(x,s) for a. a. xΩ,h2(x,s)=ζ|s|p2sg(x,s)for a.a. xΩ,

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|p2u+μ(x)|u|q2u)=θ|u|p2uμ(x)|u|q2uf(x,u) in Ω,(|u|p2u+μ(x)|u|q2u)v=ζ|u|p2ug(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:

1. f and g are bounded on bounded sets.

2. It holds

lims±f(x,s)|s|q2s=+ uniformly for a. a. xΩ,lims±g(x,s)|s|q2s=+ uniformly for a. a. xΩ.
3. It holds

lims0f(x,s)|s|q2s=0 uniformly for a. a. xΩ,lims0g(x,s)|s|p2s=0 uniformly for a.a. xΩ.

We say that uW1,ℌ(Ω) is a weak solution of problem (4.1) if

Ω(|u|p2u+μ(x)|u|q2u)φdx+Ω(ϑ|u|p2u+μ(x)|u|q2u)φdx=Ω(f(x,u))φdx+Ω(ζ|u|p2ug(x,u))φdσ

is fulfilled for all φW1,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,pSwithλ1,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, v0W1,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. xΩ and all |s|M1,g(x,s)sζ|s|q for a.a. xΩ and all |s|M2.

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

(4.3) 0f(x,u¯) for a. a. xΩ and 0ζu¯p1g(x,u¯) for a.a. xΩ.

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

0f(x,v_) for a.a. xΩ and 0ζ|v_|p2v_g(x,v_) for a.a. xΩ.

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

(4.4) θ+(x,s)={ 0 if s<0f(x,s) if 0su¯,f(x,u¯) if u¯<s θζ+(x,s)={ 0 if s<0ζsp1g(x,s) if 0su¯ζu¯p1g(x,u¯) if u¯<s ,θ(x,s)={ f(x,v_) if s<v_f(x,s) if v_s00 if 0<s ,θζ(x,s)={ ζ|_|p2v_g(x,v_) if s<v_ζ|s|p2sg(x,s) if v_s00 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± : W1,ℌ(Ω) → R defined by

Γ±(u)=1pupp+1quq,μq+ϑpupp+1quq,μqΩΘ±(x,u)dxΩΘζ±(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 u0W1,ℌ(Ω) exists, that is

Γ+(u0)=inf[ Γ+(u):uW1,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. xΩ and for all |s|δ1,G(x,s)ε2p|s|p for a.a. xΩ 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,pSint(C1(Ω¯)+), we may choose t ∈ (0, 1) small enough such that tu1,pS(x)[0,δ] for all xΩ. 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<ζλ1,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 |p2u0+μ(x)| u0 |q2u0)φdx+Ω(ϑ| u0 |p2u0+μ(x)| u0 |q2u0)φdx=Ωθ+(x,u0)φdx+Ωθζ+(x,u0)φdσ

for all φW1,ℌ(Ω). First we take φ=u0W1,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 = (u0u)+W1,ℌ(Ω) as test function in (4.7) which results in

(4.8) Ω(| u0 |p2u0+μ(x)| u0 |q2u0)(u0u¯)+dx+Ω(ϑu0p1+μ(x)u0q1)(u0u¯)+dx=Ωθ+(x,u0)(u0u¯)+dx+Ωθζ+(x,u0)(u0u¯)+dσ=Ω(f(x,u¯))(u0u¯)+dx+Ω(ζu¯p1g(x,u¯))(u0u¯)+dσ0,

by (4.3). First note that

(4.9) Ω(| u0 |p2u0+μ(x)| u0 |q2u0)(u0u¯)+dxϑΩ(| (u0u¯)+ |p+μ(x)| (u0u¯)+ |q)dx

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

(4.10) Ω(ϑu0p1+μ(x)u0q1)(u0u¯)+dxϑ{ u0>u¯ }(u0p1+μ(x)u0q1)(u0u¯)dxϑ{ u0>u¯ }((u0u¯)p1+μ(x)(u0u¯)q1)(u0u¯)dx=ϑΩ(((u0u¯)+)p+μ(x)((u0u¯)+)q)dx.

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

ϑmin{ (u0u¯)+ 0p, (u0u¯)+ 0q }ϑρ^H((u0u¯)+)0.

Hence, u0u and so u0 ∈ [0, u]. By the definition of the truncations in (4.4)we see that u0W1,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 : W1,ℌ(Ω) → R and show in the same way that it has a global minimizer v0W1,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)=(ζϑ)|s|p2sμ(x)|s|q2sf(x,s) for a.a. xΩ,h2(x,s)=β|s|p2sfor a.a. xΩ,

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|p2u+μ(x)|u|q2u)=(ζϑ)|u|p2uμ(x)|u|q2uf(x,u) in Ω,(|u|p2u+μ(x)|u|q2u)v=β|u|p2uon Ω,

where f satisfies the following assumptions:

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

1. f is bounded on bounded sets.

2. It holds

lims±f(x,s)|s|q2s=+ uniformly for a. a. xΩ.
3. It holds

lims0f(x,s)|s|p2s=0 uniformly for a. a. xΩ.

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, v1W1,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)sζ|s|q for a.a. xΩ 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) 0ζu¯p1f(x,u¯) for a. a. xΩ and 0ζ|v_|p2v_f(x,v_) for a.a. xΩ,

because p < q and M > 1.

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

(4.14) ψζ+(x,s)={ 0 if s<0ζsp1f(x,s) if 0su¯ζu¯p1f(x,u¯) if u¯<s ,ψβ+(x,s)={ 0 if s<0βsp1 if 0su¯βu¯p1 if u¯<s ,ψζ(x,s)={ ζ|v_|p2v_f(x,v_) if s<v_ζ|s|p2sf(x,s) if v_s00 if 0<s ,ψβ(x,s)={ β|v_|p2v_ if s<v_β|s|p2s if v_s00 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± : W1,H(Ω) → R given by

Π±(u)=1pupp+1quq,μq+ϑpupp+1quq,μqΩΨζ±(x,u)dxΩΨβ±(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 u1W1,H(Ω) such that

Π+(u1)=inf[ Π+(u):uW1,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. xΩ 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,βRint(C1(Ω¯)+) we choose t ∈ (0, 1) sufficiently small such that tu1,p,βR(x)[0,δ] for all xΩ¯. 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,ζλ1,p,βRϑ) 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 |p2u1+μ(x)| u1 |q2u1)φdx+Ω(θ| u1 |p2u1+μ(x)| u1 |q2u1)φdx=Ωψζ+(x,u1)φdx+Ωψβ+(x,u1)φdσ

for all φ W1,ℌ(Ω). As done in the proof of Theorem 4.1 we take φ = −u1W1,ℌ(Ω) and φ = (u1u)+W1,ℌ(Ω) as test functions in (4.17) which gives us 0 ≤ u1u, see (4.13). Hence, by the definition of the truncations in (4.14) we see that u1W1,ℌ(Ω) ∩ L(Ω) solves problem (4.11).

In the same way we can show the existence of a nontrivial nonpositive solution v1W1,ℌ(Ω) ∩ L(Ω) by treating the functional Π : W1,ℌ(Ω) → R instead of Π+ : W1,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 > λS1,p by (4.18) where u1,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.

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.

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

References

[1] A. Bahrouni, V.D. Rădulescu, P. Winkert, Double phase problems with variable growth and convection for the Baouendi-Grushin operator, Z. Angew. Math. Phys. 71 (2020), no. 6, 183, 14 pp.Search in Google Scholar

[2] G. Barletta, E. Tornatore, Elliptic problems with convection terms in Orlicz spaces, J. Math. Anal. Appl. 495 (2021), no. 2, 124779, 28 pp.Search in Google Scholar

[3] P. Baroni, M. Colombo, G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal. 121 (2015), 206–222.Search in Google Scholar

[4] P. Baroni, M. Colombo, G. Mingione, Non-autonomous functionals, borderline cases and related function classes, St. Petersburg Math. J. 27 (2016), 347–379.Search in Google Scholar

[5] P. Baroni, M. Colombo, G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations 57 (2018), no. 2, Art. 62, 48 pp.Search in Google Scholar

[6] P. Baroni, T. Kuusi, G. Mingione, Borderline gradient continuity of minima, J. Fixed Point Theory Appl. 15 (2014), no. 2, 537–575.Search in Google Scholar

[7] S.-S. Byun, J. Oh, Regularity results for generalized double phase functionals, Anal. PDE 13 (2020), no. 5, 1269–1300.Search in Google Scholar

[8] F. Colasuonno, M. Squassina, Eigenvalues for double phase variational integrals, Ann. Mat. Pura Appl. (4) 195 (2016), no. 6, 1917–1959.Search in Google Scholar

[9] M. Colombo, G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal. 218 (2015), no. 1, 219–273.Search in Google Scholar

[10] M. Colombo, G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal. 215 (2015), no. 2, 443–496.Search in Google Scholar

[11] Á. Crespo-Blanco, L. Gasiński, P. Harjulehto, P. Winkert, A new class of double phase variable exponent problems: Existence and uniqueness, preprint 2021, arXiv: 2103.08928.Search in Google Scholar

[12] C. Farkas, P. Winkert, An existence result for singular Finsler double phase problems, J. Differential Equations 286 (2021), 455–473.Search in Google Scholar

[13] C. Farkas, A. Fiscella, P. Winkert, Singular Finsler double phase problems with nonlinear boundary condition, preprint 2021, arXiv: 2102.05467.Search in Google Scholar

[14] L. Gasiński, N.S. Papageorgiou, Constant sign and nodal solutions for superlinear double phase problems, Adv. Calc. Var., https://doi.org/10.1515/acv-2019-0040Search in Google Scholar

[15] L. Gasiński, P. Winkert, Constant sign solutions for double phase problems with superlinear nonlinearity, Nonlinear Anal. 195 (2020), 111739, 9 pp.Search in Google Scholar

[16] L. Gasiński, P. Winkert, Existence and uniqueness results for double phase problems with convection term, J. Differential Equations 268 (2020), no. 8, 4183–4193.Search in Google Scholar

[17] L. Gasiński, P. Winkert, Sign changing solution for a double phase problem with nonlinear boundary condition via the Nehari manifold, J. Differential Equations 274 (2021), 1037–1066.Search in Google Scholar

[18] A. Lê, Eigenvalue problems for the p-Laplacian, Nonlinear Anal. 64 (2006), no. 5, 1057–1099.Search in Google Scholar

[19] W. Liu, G. Dai, Existence and multiplicity results for double phase problem, J. Differential Equations 265 (2018), no. 9, 4311–4334.Search in Google Scholar

[20] W. Liu, G. Dai, Multiplicity results for double phase problems inN, J. Math. Phys. 61 (2020), no. 9, 091508, 20 pp.Search in Google Scholar

[21] P. Marcellini, Regularity and existence of solutions of elliptic equations with p, q-growth conditions, J. Differential Equations 90 (1991), no. 1, 1–30.Search in Google Scholar

[22] P. Marcellini, The stored-energy for some discontinuous deformations in nonlinear elasticity, in “Partial differential equations and the calculus of variations, Vol. II”, vol. 2, 767–786, Birkhäuser Boston, Boston, 1989.Search in Google Scholar

[23] G. Marino, P. Winkert, Existence and uniqueness of elliptic systems with double phase operators and convection terms, J. Math. Anal. Appl. 492 (2020), 124423, 13 pp.Search in Google Scholar

[24] G. Mingione, V.D. Rădulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, J. Math. Anal. Appl. 501 (2021), no. 1, 125197, 41 pp.Search in Google Scholar

[25] J. Ok, Partial regularity for general systems of double phase type with continuous coefficients, Nonlinear Anal. 177 (2018), 673–698.Search in Google Scholar

[26] J. Ok, Regularity for double phase problems under additional integrability assumptions, Nonlinear Anal. 194 (2020), 111408.Search in Google Scholar

[27] N.S. Papageorgiou, V.D. Rădulescu, D.D. Repovš, Double-phase problems and a discontinuity property of the spectrum, Proc. Amer. Math. Soc. 147 (2019), no. 7, 2899–2910.Search in Google Scholar

[28] N.S. Papageorgiou, V.D. Rădulescu, D.D. Repovš, Existence and multiplicity of solutions for double-phase Robin problems, Bull. Lond. Math. Soc. 52 (2020), no. 3, 546–560.Search in Google Scholar

[29] N.S. Papageorgiou, C. Vetro, F. Vetro, Solutions for parametric double phase Robin problems, Asymptot. Anal. 121 (2021), no. 2, 159–170.Search in Google Scholar

[30] N.S. Papageorgiou, P. Winkert, “Applied Nonlinear Functional Analysis. An Introduction”, De Gruyter, Berlin, 2018.Search in Google Scholar

[31] K. Perera, M. Squassina, Existence results for double-phase problems via Morse theory, Commun. Contemp. Math. 20 (2018), no. 2, 1750023, 14 pp.Search in Google Scholar

[32] V.D. Rădulescu, Isotropic and anistropic double-phase problems: old and new, Opuscula Math. 39 (2019), no. 2, 259–279.Search in Google Scholar

[33] M.A. Ragusa, A. Tachikawa, Regularity for minimizers for functionals of double phase with variable exponents, Adv. Nonlinear Anal. 9 (2020), no. 1, 710–728.Search in Google Scholar

[34] S.D. Zeng, Y.R. Bai, L. Gasiński, P. Winkert, Convergence analysis for double phase obstacle problems with multivalued convection term, Adv. Nonlinear Anal. 10 (2021), no. 1, 659–672.Search in Google Scholar

[35] S.D. Zeng, Y.R. Bai, L. Gasiński, P. Winkert, Existence results for double phase implicit obstacle problems involving multivalued operators, Calc. Var. Partial Differential Equations 59 (2020), no. 5, 176.Search in Google Scholar

[36] S.D. Zeng, L. Gasiński, P. Winkert, Y.R. Bai, Existence of solutions for double phase obstacle problems with multivalued convection term, J. Math. Anal. Appl. 501 (2021), no. 1, 123997, 12 pp.Search in Google Scholar

[37] V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 675–710.Search in Google Scholar

[38] V.V. Zhikov, S. M. Kozlov, O. A. Ole˘ınik, “Homogenization of Differential Operators and Integral Functionals”, Springer-Verlag, Berlin, 1994.Search in Google Scholar