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

: 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.


Introduction
Let Ω ⊂ R N , N > , be a bounded domain with Lipschitz boundary ∂Ω. We consider the following double phase problem with nonlinear boundary condition and convection term given by − div |∇u| p− ∇u + µ(x)|∇u| q− ∇u = h (x, u, ∇u) in Ω, where ν(x) is the outer unit normal of Ω at the point x ∈ ∂Ω, < p < q < N, ≤ µ(·) ∈ L (Ω) and h : Ω × R × R N → R as well as h : ∂Ω × R → R are Carathéodory functions which satisfy suitable structure conditions and behaviors near the origin and at in nity, see Sections 3 and 4 for the precise assumptions. The di erential operator that appears in (1.1) is the so-called double phase operator which is de ned by − div |∇u| p− ∇u + µ(x)|∇u| q− ∇u for u ∈ W ,H (Ω) (1.2) with an appropriate Musielak-Orlicz Sobolev space W ,H (Ω), see its de nition in Section 2. Note that when inf Ω µ > or µ ≡ then the operator becomes the weighted (q, p)-Laplacian or the p-Laplacian, respectively.
The energy functional J : W ,H (Ω) → R related to the double phase operator (1.2) is given by (1.3) 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) = }. 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 rst 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 paper we are going to study problem (1.1) concerning multiplicity of solutions. In the rst part of the paper, see Section 3, we prove the existence of a nontrivial weak solution when the function h 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 operators where in the proof the rst 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 rst 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 in Ω, where < q < p < N, ξ ∈ L ∞ (Ω) is a positive potential, a(z) > for a. a. z ∈ Ω and ∂u ∂n θ = [a(z)|∇u| p− + |∇u| q− ] ∂u ∂n with n(·) being the outward unit normal on ∂Ω. Under di erent assumptions it is shown that problem (1.4) admits two nontrivial solutions u λ ,û λ ∈ W ,H (Ω) for small λ > such that u λ ,H → +∞ and û λ ,H → as λ → + . In Papageorgiou-Rădulescu-Repovš [28] the authors proved the existence of multiple solutions in the superlinear and the resonant case for the problem where < q < p ≤ N and with a positive Lipschitz function a (·). Note that our assumptions and our treatment di er 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) 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 di erent 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 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. Related works 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 ,H (Ω). In Section 3 we prove the existence of at least one solution of (1.1) when h 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 di erent problems. The rst problem is treated by properties of the rst Steklov eigenvalue and the second one by the rst Robin eigenvalue, see Theorems 4.1 and 4.2.

Preliminaries
In this section we recall some de nitions and present the main tools which will be needed in the sequel. For every ≤ r < ∞ we denote by L r (Ω) and L r (Ω; R N ) the usual Lebesgue spaces equipped with the norm · r and for < r < ∞ we consider the corresponding Sobolev space W ,r (Ω) endowed with the norm · ,r . It is known that W ,r (Ω) → Lr(Ω) is compact forr < r * and continuous forr = r * , where r * is the critical exponent of r de ned by (2.1) On the boundary ∂Ω of Ω we consider the (N − )-dimensional Hausdor (surface) measure σ and denote by L r (∂Ω) the boundary Lebesgue space with norm · r,∂Ω . From the de nition of the trace mapping we know that W ,r (Ω) → Lr(∂Ω) is compact forr < r * and continuous forr = r * , where r * is the critical exponent of r on the boundary given by For simpli cation we will avoid the notation of the trace operator throughout the paper.
In the entire paper we will assume that 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 is needed, which is equivalent to q < p * and so q < p * is also satis ed. We do not need this restriction in the current paper.
Based on this we can introduce the modular function given by From Colasuonno-Squassina [8, Proposition 2.14] we know that the space L H (Ω) is a re exive Banach space. Moreover, we need the seminormed space which is endowed with the seminorm where ∇u H = |∇u| H . As before, we know that W ,H (Ω) is a re exive Banach space.

Proposition 2.1. Let (2.3) be satis ed and let
be the critical exponents to p, see (2.1) and (2.2) for r = p. Then the following embeddings hold: We equip the space W ,H (Ω) with the equivalent norm Let us recall some de nitions which we will need in the next sections.

De nition 2.3.
Let (X, · X ) be a re exive Banach space, X * its dual space and denote by · , · its duality pairing. Let A : X → X * , then A is called Remark 2.4. The classical de nition of pseudomonotonicity is the following one: From un u in X and lim sup n→∞ Aun , un − u ≤ we have This de nition is equivalent to the one in De nition 2.3(ii) when the operator is bounded. Since we are only considering bounded operators, we will use the one in De nition 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, re exive 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 ,H (Ω) → W ,H (Ω) * be the nonlinear map de ned by

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, } and for u ∈ W ,H (Ω) we de ne u ± (·) = u(·) ± . We have Let us now recall some basic facts about the spectrum of the negative r-Laplacian with Robin and Steklov boundary condition, respectively, for < r < ∞. We refer to the paper of Lê [18]. The r-Laplacian eigenvalue problem with Robin boundary condition is given by where β > . We know that problem (2.7) has a smallest eigenvalue λ R ,r,β > which is isolated, simple and it can be variationally characterized by (2.8) By u R ,r,β we denote the normalized (that is, u R ,r,β r = ) positive eigenfunction corresponding to λ R ,r,β . We know that u R ,r,β ∈ int C (Ω)+ . Further, we recall the r-Laplacian eigenvalue problem with Steklov boundary condition which is given by −∆r u = −|u| r− u in Ω, As before, problem (2.9) has a smallest eigenvalue λ S ,r > which is isolated, simple and which can be characterized by (2.10) The rst eigenfunction associated to the rst eigenvalue λ S ,r will be denoted by u S ,r and we can assume it is normalized, that is, u S ,r r,∂Ω = . We have u S ,r ∈ int C (Ω)+ .

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 rst eigenvalues of the Robin and Steklov eigenvalue problems of the p-Laplacian. We choose for all s ∈ R and for all ξ ∈ R N with ζ > speci ed later and Carathéodory functions f and g characterized in hypotheses (H1) below. Then (1.1) becomes 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, , ) ≠ for a. a.
x ∈ Ω such that the following conditions are satis ed: (i) There exist α ∈ L r r − (Ω), α ∈ L r r − (∂Ω) and a , a , a ≥ such that for all s ∈ R and for all ξ ∈ R N , where < r < p * and < r < p * with the critical exponents p * and p * stated in (2.4). (ii) There exist w ∈ L (Ω), w ∈ L (∂Ω) and b , b , b ≥ such that for all s ∈ R and for all ξ ∈ R N .
A function u ∈ W ,H (Ω) is called a weak solution of problem (3.1) if is satis ed for all φ ∈ W ,H (Ω). It is clear that this de nition is well-de ned.
The main result in this section is the following one.  Applying the growth conditions in (H1)(i) along with Hölder's inequality gives This shows the coercivity of A. As before, let u ∈ W ,H (Ω) be such that u > and note again that W ,H (Ω) ⊆ W ,p (Ω). Applying (H1)(ii), (3.6), (B) and Proposition 2.2(iv) one gets Hence, A : W ,H (Ω) → W ,H (Ω) * is again coercive. We have shown that A : W ,H (Ω) → W ,H (Ω) * is a bounded, pseudomonotone and coercive operator. From Theorem 2.5 we nd an elementû ∈ W ,H (Ω) such that A(û) = withû ≠ since f (x, , ) ≠ for a. a. x ∈ Ω. In view of the de nition of A, we see thatû 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 ofû. The proof is complete.

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 for all s ∈ R, ϑ, ζ > to be speci ed and Carathéodory functions f and g which satisfy hypotheses (H2) below. With this choice, (1.1) can be written as 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: (i) f and g are bounded on bounded sets. We say that u ∈ W ,H (Ω) is a weak solution of problem (4.1) if is ful lled for all φ ∈ W ,H (Ω).
The following theorem states the existence of constant sign solutions where the parameter ζ depends on the rst Steklov eigenvalue for the p-Laplacian, namely λ S ,p . Analogously, we can choose v ≡ −ς in order to get Now, we introduce the cut-o functions θ ± : Ω × R → R and θ ± ζ : ∂Ω × R → R de ned by which are Carathéodory functions. We set Now we consider the C -functionals Γ ± : W ,H (Ω) → R de ned by Furthermore, we write F(x, s) = s f (x, t) dt and G(x, s) = s g(x, t) dt.
We rst 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 u ∈ W ,H (Ω) exists, that is Γ + (u ) = inf Γ + (u) : u ∈ W ,H (Ω) .
From this choice and since p < q we obtain from (4.6) Γ + tu S ,p < for all su ciently small t > .
Therefore, we know now that Hence, u ≠ . Since u is a global minimizer of Γ + we have (Γ + ) ′ (u ) = , that is, for all φ ∈ W ,H (Ω). First we take φ = −u − ∈ W ,H (Ω) as test function in (4.7). We obtain which yields u − = and so u ≥ . Second we choose φ = (u − u) + ∈ W ,H (Ω) as test function in (4.7) which results in (4.9) Since u > u > on the set {u > u} we have (4.10) Combining (4.8) with (4.9) as well as (4.10) and using Proposition 2.2(iii), (iv) implies that Hence, u ≤ u and so u ∈ [ , u]. By the de nition of the truncations in (4.4) we see that u ∈ W ,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 ,H (Ω) → R and show in the same way that it has a global minimizer v ∈ W ,H (Ω) which belongs to [v, ].
We have the following multiplicity result concerning problem (4.11). because p < q and M > . Then we de ne truncations ψ ± ζ : Ω × R → R and ψ ± β : ∂Ω × R → R as follows if < s (4.14) We set which depends also on the boundary norm of the eigenfunction u R ,p,β . So the statement of Theorem 4.1 still holds true when we replace the assumption ζ > λ S ,p by (4.18) where u R ,p,β is the rst normalized (that is, u R ,p,β p = ) eigenfunction associated to the rst eigenvalue λ R ,p,β of the Robin eigenvalue problem.