Existence of solutions for resonant double phase problems with mixed boundary value conditions

We study a double phase problem with mixed boundary value conditions with reaction terms that resonate at the first eigenvalue of the related eigenvalue problem. Based on the maximum principle and homological local linking, we are going to prove the existence of at least two bounded nontrivial solutions for this problem.


Introduction
In this paper, we study the following double phase problems with mixed boundary conditions A(u) + |u| p−2 u + a(x) |u| q−2 u = f (x, u) in , |∇u| p−2 ∇u + a(x)|∇u| q−2 ∇u · ν = g(x, u) on , This article is part of the section "Theory of PDEs" edited by Eduardo Teixeira.  1 where A(u) := − div |∇u| p−2 ∇u + a(x)|∇u| q−2 ∇u is the double phase operator, is a bounded domain of R N , N ≥ 2, with a C 1 boundary ∂ such that ∂ = σ ∪ and σ ∩ = ∅, ν(x) denotes the outer unit normal of at x ∈ , Clearly, q < p * implies q < p * = N p N − p . The nonlinearities f and g satisfy the following hypotheses: (H) f : × R → R and g : × R → R are Carathéodory functions such that the following hold: (i) There exist constants C 1 , C 2 > 0 such that |g(x, t)| ≤ C 2 1 + |t| r 2 −1 for a.a. x ∈ , for all t ∈ R, where q < r 1 < p * and q < r 2 < p * , respectively. (iv) There exist δ > 0, θ >λ 1 ( p) and 0 <λ <λ 2 ( p) such that θ |t| p ≤ pF(x, t) ≤λ |t| p for a.a. x ∈ and for all |t| ≤ δ, θ |t| p ≤ pG(x, t) ≤λ |t| p for a.a. x ∈ and for all |t| ≤ δ, where λ 1 (q) stands for the first eigenvalue of the weighted q-Laplace mixed boundary condition problem whileλ 1 ( p) andλ 2 ( p) represent the first and the second eigenvalues of the p-Laplace mixed boundary condition problem, respectively, see Sect. 2 for more details.
The solutions of problem (1.1) are understood in the weak sense, that is, u ∈ X is a solution of (1.1) if  [38,39] and the references therein. The purpose of this paper is to study the multiplicity of solutions for problem (1.1). There are two main characteristics of this problem: one is that the reaction terms resonate at the corresponding eigenvalues; the other one is the appearance of nonlinear boundary conditions and mixed boundary conditions.
The main result in this paper is the following theorem.  [31] investigated the existence of multiple solutions to a double phase Robin problem when resonating at the first eigenvalue of the weighted p-Laplace Robin problem, applying the local linking of the Morse theory to derive the existence of at least two bounded solutions. Papageorgiou-Rȃdulescu-Zhang [33] considered the existence of multiple solutions to the Dirichlet double phase problem when resonating at the first eigenvalue of the weighted p-Laplace Dirichlet equation, using variational methods together with Morse theory to yield the existence of at least two bounded nontrivial solutions. Liu-Zeng-Gasiński-Kim [23] studied a nonlinear complementarity problem (NCP) with a double phase differential operator and a generalized multivalued boundary condition. By using the Moreau-Yosida approximation method, the regularization problem corresponding to NCP was introduced, and finally, the properties of the solution set of NCP were obtained. Inspired by the above papers, we are going to study the resonant double phase equations under mixed boundary conditions given in (1.1) in the present paper. The reaction terms resonate at the first eigenvalue of the weighted q-Laplace equation with the mixed boundary, which is different from Papageorgiou-Rȃdulescu-Repovš [31] and Papageorgiou-Rȃdulescu-Zhang [33].
The mixed boundary conditions are divided into two parts, one is the Dirichlet boundary condition and the other is the nonlinear boundary condition, which is different from Liu-Zeng-Gasiński-Kim [23]. These differences bring new challenges. In order to overcome these difficulties, we need to require more elaborate calculations to get the compactness condition and the homological local linking.
The proof of Theorem 1.1 is based on variational methods and Morse theoretic aspects, especially the homological local linking. First, by the hypotheses (H)(i) and (H)(iii), we show that the corresponding energy functional J of (1.1) satisfies the Cerami condition. Second, by (H)(ii) and (H)(iii), we prove that J is coercive, and then by the Weierstrass-Tonelli theorem, it is concluded that there exists u 1 = 0 such that J (u 1 ) = 0. Finally, in order to obtain the second solution u 2 , we verify that J has a local (1, 1)-linking at 0 by hypothesis (H)(ii). In addition, we study the eigenvalue problem of the weighted q-Laplace equation with mixed boundary conditions. The rest of this paper is organized as follows. In Sect. 2 we recall some main variational tools and introduce the Musielak-Orlicz spaces L H ( ) and W 1,H ( ) including some of its properties. We also present some properties of the weighted q-Laplace equation with mixed boundary conditions and the related first eigenvalue and its eigenfunction. The proof of the Theorem 1.1 is then given in Sect. 3.
Suppose (1.2) and let H : Then, the Musielak-Orlicz space L H ( ) is defined by where the modular function ρ H (·) is given by We know that the space L H ( ) is a reflexive Banach space. Moreover, we define the weighted which is endowed with the seminorm It is not easy to check the validity of the following continuous embeddings Similarly, we define which is endowed with the norm We know that W Then the following embedding hold: The norm · and the modular function are related as follows, see Liu Let L : W 1,H ( ) → W 1,H ( ) * be the nonlinear operator given by Next, we recall some definitions and tools that will be used in this paper.

Definition 2.4
Let X be a real Banach space and let X * be its dual space. We say that J ∈ C 1 (X ) satisfies the Cerami-condition (C-condition for short), if for any {u n } n∈N ⊆ X such that {J (u n )} n∈N ⊆ R is bounded and (1 + u n )J (u n ) → 0 in X * , admits a strongly convergent subsequence.

Proposition 2.5
Suppose that X is a reflexive Banach space. If J : X → R is coercive and sequentially weakly lower semi-continuous on X , then I is bounded from below on X and has a minimum in X .
Let X be a Banach space, J ∈ C 1 (X , R) and c ∈ R. We introduce the following sets Consider a topological pair (A, B) such that B ⊆ A ⊆ X . For every k ∈ N 0 we denote by H k (A, B) the kth-relative singular homology group with integer coefficients for the pair (A, B). If u ∈ K c J is isolated, the critical groups of J at u are defined by The excision property of singular homology implies that this definition is independent of the choice of the isolating neighborhood U .
If J ∈ C 1 (X , R) satisfies the C-condition (see Definition 2.4), inf J (K J ) > −∞ and c < inf J (K J ), then the critical groups of J at infinity are defined by Taking Corollary 5.3.12 of Papageorgiou-Rȃdulescu-Repovš [32] into account, this definition is independent of the choice of the level c < inf J (K J ).
We use the local (m, n)-linking method to prove the existence of a solution of problem (1.1). The following definition is originally due to Perera [35] (see also Papageorgiou-Rȃdulescu-Repovš [32, Definition 6.6.13]).

Definition 2.6
Let X be a Banach space, J ∈ C 1 (X , R), and 0 an isolated critical point of Next, we want to study an appropriate eigenvalue problem following the ideas of Papageorgiou-Rȃdulescu-Repovš [31] and Li-Liu-Cheng [17]. We consider the following weighted q-Laplacian eigenvalue problem with mixed boundary conditions where q and a(·) satisfy the hypothesis (1.2). The first eigenvalue λ 1 (q) > 0 of (2.3) has the following variational characterization The corresponding eigenfunction u 1 ∈ W 1,K ( ) to the first eigenvalue λ 1 > 0 satisfies u 1 ∈ L ∞ ( ) and u 1 (x) > 0 for a.a. x ∈ which can be shown similar to Proposition 3 of Papageorgiou-Rȃdulescu-Zhang [33]. Furthermore, letλ 1 ( p) be the first eigenvalue of the following p-Laplacian mixed boundary value problem − div |∇u| p−2 ∇u = λ|u| p−2 u in , (2.5) Based on the results of Li-Liu-Cheng [17] we know that the first eigenvalueλ 1 ( p) of (2.5) is positive, simple and isolate. Letũ 1 be the positive eigenfunction associated withλ 1 ( p), thenũ 1 ∈ L ∞ ( ). Moreover, the second eigenvalueλ 2 ( p) of (2.5) can be written as

Proof of Theorem 1.1
In this section, we are going to prove Theorem 1.1. Recall that X = {u ∈ W 1,H ( ) : u| σ = 0} and let u = u 1,H for all u ∈ X be the norm of X . The corresponding energy functional J : X → R related to problem (1.1) is given by Under our assumptions, it is standard to check that J : X → R is well-defined and of class C 1 and the solutions of problem (1.1) are the critical points of J : X → R. First, we will show that J : X → R satisfies the C-condition. and By (3.2) and (2.2), we have for all v ∈ X with ε n → 0 + , which implies that Taking v = u n in (3.3), it follows from (3.3) and (3.4) that, for all n ∈ N. Moreover, by (3.1) we obtain Adding (3.5) and (3.6) we have for some M 2 > 0. Since p < q, we get in particular that Suppose that u n → ∞. We take v n = u n u n which implies that v n = 1. Then we may assume that v n v in X and v n → v in L r 1 ( ) and L r 2 (∂ ) for some v ∈ X , see Proposition 2.1(ii), (iv).
Suppose v = 0. Let μ ≥ 1 and putṽ n = (qμ) 1 q v n for all n ∈ N. So we haveṽ n → 0 in L r 1 ( ) and L r 2 (∂ ), which implies that Thus, for all ε > 0 we can find n 0 ∈ N such that for all n ≥ n 0 . Now we choose t n ∈ [0, 1] such that Recalling u n → ∞ as n → ∞, we can find n 1 ∈ N such that 0 < (qμ) 1 q u n ≤ 1 for all n ≥ n 1 . (3.11) Taking ε = 1 2 min q p q p , 1 μ p q in (3.9), we conclude from (3.9), (3.10) and (3.11) that for all n ≥ max{n 1 , n 0 }. Since μ ≥ 1 is arbitrary, we obtain J (t n u n ) → +∞ as n → ∞. f (x, t n u n )t n u n dx + g(x, t n u n )t n u n dS (3.14) for all n ≥ n 2 . Hence, from (3.14) we have t n u n )u n − q F(x, t n u n )) dx + (g(x, t n u n )u n − qG(x, t n u n )) dS for all n ≥ n 2 . It follows from (3.7) that qϕ(t n u n ) ≤ M 2 for all n ≥ n 2 .
which contradicts (3.12). Suppose now v ≡ 0. Letˆ = ∪ and definê Then at least one of these measurable sets has a positive Lebesgue measure on R N . Note that u n (x) → +∞ for a.a. x ∈ˆ + and u n (x) → −∞ for a.a. x ∈ˆ − .
From the boundedness of the sequence, we can find a subsequence, still denoted by {u n } n∈N , such that u n u in X and u n → u in L r 1 ( ) and in L r 2 (∂ ).
We choose v = u n − u in (3.3) and obtain using the convergence properties above that lim n→∞ L(u n ), u n − u H = 0.
Therefore, it follows that u n → u in X since L is a mapping of type (S + ), see Proposition 2.3.
Next, we prove that J : X → R is coercive. (1.2) and (H) be satisfied, then the energy functional J : X → R is coercive.
Hence, we obtain d dt Integrating this inequality, we obtain for a.a. x ∈ and for all |t| ≥ |u| ≥ M δ . By hypothesis (H)(ii), for any ε > 0, there exists M ε > 0 such that a. x ∈ and for all |t| ≥ M ε .
Using this inequality in (3.15) and letting |t| → ∞, we obtain for a.a. x ∈ and for all |u| ≥ M = max{M δ , M ε }. Letting ε → 0, we obtain (3.16) Similar arguments apply to G(·, ·), that is, we can show for a.a. x ∈ and for all |u| ≥ M. Now, we claim that J : X → R is coercive. Indeed, for any u ∈ X , it follows from X ⊂ W 1,H ( ) → W 1,K , (2.4) and (3.16) as well as (3.17) that for a.a. x ∈ ∪ and for all |u| ≥ M, which implies that J : X → R is coercive since δ is arbitrary and | | , | | > 0.
Finally, we will prove that J : X → R has a local (1, 1)-linking at 0. Proof Let V denote the space spanned byũ 1 ( p) and let We claim that Indeed, for any u ∈ X , writing u = αũ 1 + w where w ∈ X and Recall thatλ see (2.5). Thus we obtain Hence, w ∈ W and our claim is true. We may assume that K J is finite, otherwise we would have found infinite number of critical points of J which are solutions of problem (1.1). Now, let B ρ = {u ∈ X : u ≤ ρ} and choose ρ ∈ (0, 1) small enough such that K J ∩ B ρ = {0}. Furthermore, let ε > 0 small enough such that the hypothesis (H)(iv) holds, that is, Then for tũ 1 = u ∈ V ∩ B ρ with t ∈ (0, 1), by (3.19), (3.20) and (3.21) we have Taking ρ ∈ (0, 1) small enough yields Then we obtain that J : X → R has a local (1, 1)-linking at 0, see Definition 2.6.
Based on the results above, we are now in the position to prove Theorem 1.1.
Proof of Theorem 1.1 First, since W 1,H ( ) → L r 1 ( ) and W 1,H ( ) → L r 2 (∂ ) are compact due to Proposition 2.1, we know that J : X → R is sequentially weakly lower semicontinuous. From Proposition 3.2 we conclude that J : X → R is coercive as well. Therefore, by Proposition 2.5, we deduce that there exists u 1 ∈ X such that J (u 1 ) = min{J (u) : u ∈ X }.
By the proof of Proposition 3.3, we see that J (u 1 ) < 0 = J (0), which implies that u 1 = 0 and u 1 ∈ K J , that is, u 1 ∈ K J is a nontrivial solution of problem (1.1). Moreover, it follows from Propositions 2.7, 3.1 and 3.3 that there exists u 2 ∈ K J such that u 2 / ∈ {0, u 1 }, which implies that u 2 is the second nontrivial solution of problem (1.1). From Theorem 3.1 of Gasiński-Winkert [15] we conclude that u 1 and u 2 are bounded.