Bounded weak solutions to superlinear Dirichlet double phase problems

In this paper we study a Dirichlet double phase problem with a parametric superlinear right-hand side that has subcritical growth. Under very general assumptions on the data, we prove the existence of at least two nontrivial bounded weak solutions to such problem by using variational methods and critical point theory. In contrast to other works we do not need to suppose the Ambrosetti–Rabinowitz condition.


Introduction
In this paper we consider the following Dirichlet double phase problem − div |∇u| p−2 ∇u + μ(x)|∇u| q−2 ∇u = λ f (x, u) in , where ⊆ R N , N ≥ 2, is a bounded domain with Lipschitz boundary ∂ , 1 < p < N , p < q < p * and 0 ≤ μ(·) ∈ L ∞ ( ) with p * = N p N − p , λ > 0 is a parameter and f : × R → R is a Carathéodory function that satisfies subcritical growth and a certain behavior at ±∞.
The operator involved is the so-called double phase operator defined by div |∇u| p−2 ∇u + μ(x)|∇u| q−2 ∇u for u ∈ W 1,H 0 ( ) (1.2) with W 1,H 0 ( ) being an appropriate Musielak-Orlicz Sobolev space, see its Definition in Sect. 2. It is clear that (1.2) reduces to the p-Laplacian if μ ≡ 0 and to the (q, p)-Laplacian if inf μ ≥ μ 0 > 0. Moreover, the double phase operator is related to the two-phase integral functional J : W 1,H 0 ( ) → R given by Zhikov [24] was the first who introduced and studied functionals of type (1.3) whose integrands change their ellipticity according to a point in order to provide models for strongly anisotropic materials. It is clear that the integrand of (1.3) has unbalanced growth. The main characteristic of (1.3) is the change of ellipticity on the set where the weight function is zero, that is, on the set {x ∈ : μ(x) = 0}. In other words, the energy density of (1.3) 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. We also refer to the book of Zhikov-Kozlov-Oleȋnik [25]. Functionals of the form (1.3) have been studied by several authors with respect to regularity of local minimizers, see, for example, the works of Baroni-Colombo-Mingione [1][2][3], Colombo-Mingione [9,10] and for nonautonomous integrals, the recent work of De Filippis-Mingione [12]. The main objective of our work is to apply an abstract critical point theorem to problem (1.1) in order to get two nontrivial bounded weak solutions with different energy sign. In addition, we give a precise interval to which the solutions belong. Our paper can be seen as an extension of a work of the first two authors recently published in [22]. The differences to [22] are twofold: First, in [22] the operator is the well-known (q, p)-Laplacian and so the function space is a usual Sobolev space. Second, we are able to weaken the assumptions on f in our paper. Indeed, in contrast to [22] and lots of other works in this direction we do not need to assume that f fulfills the usual Ambrosetti-Rabinowitz condition, which says, that there existμ > q and M > 0 such that for a. a. x ∈ and for all |s| ≥ M. Instead of (AR) we suppose that the primitive of f is q-superlinear at ±∞ (see (H2)(ii)) and we have another behavior near ±∞, see (H2)(iii). Both conditions are weaker than (AR) and they also imply that f is (q − 1)superlinear at ±∞. Note that we do not need any behavior of f or its primitive near the origin, see Theorem 3.4.
The paper is organized as follows. In Sect. 2 we recall some facts about Musielak-Orlicz Sobolev spaces and state the abstract critical point theorem mentioned above, see Theorem 2.4. Then, in Sect. 3 we formulate our hypotheses, state and prove our main result, see Theorem 3.4 and we consider some consequences for special cases of (1.1), see Corollaries 3.5 and 3.6, especially when f is nonnegative and independent of x.

Preliminaries
In this section we recall some preliminary facts and tools which are needed in the sequel. To this end, let ⊆ R N , N ≥ 2 be a bounded domain with Lipschitz boundary ∂ . For 1 ≤ r ≤ ∞ we denote by L r ( ) and L r ( ; R N ) the usual Lebesgue spaces equipped with the norm · r and for 1 ≤ r < ∞, W 1,r ( ) and W 1,r 0 ( ) stand for the Sobolev spaces endowed with the norms · 1,r and · 1,r ,0 = ∇ · r , respectively. Let 1 < p < ∞. From the Sobolev embedding theorem we know that for any ∈ [1, p * ] we have the continuous embedding W It is clear that the embedding in (2.1) is compact if < p * . Suppose that < p * . Then, from Hölder's inequality and (2.1), we obtain with | | being the Lebesgue measure of in R N .
Then we can find an element x 0 ∈ such that the ball with center x 0 and radius R > 0 belongs to , that is, We set In the following we use the subsequent assumptions: Then, the Musielak-Orlicz space L H ( ) is defined by where the modular function ρ H is given by Furthermore, we define the seminormed space which is endowed with the seminorm

Proposition 2.1 Let (H1) be satisfied, let y ∈ L H ( ) and let ρ H be defined by (2.7).
Then the following hold: We have the following embedding results for the spaces L H ( ) and W 1,H 0 ( ), see [11,Proposition 2.16].
Let X be a Banach space and X * its topological dual space. Given ϕ ∈ C 1 (X ) we say that ϕ satisfies the Cerami-condition (C-condition for short), if every sequence admits a strongly convergent subsequence.
The following theorem is used in our proofs and can be found in the paper of Bonanno-D'Aguì [4, see Theorem 2.1 and Remark 2.2]. Theorem 2.4 Let X be a real Banach space and let , : X → R be two functionals of class C 1 such that inf X (u) = (0) = (0) = 0. Assume that there are r ∈ R and u ∈ X , with 0 < (ũ) < r , such that

9)
and, for each the functional I λ = − λ satisfies the C-condition and it is unbounded from below. Moreover, is supposed to be coercive. Then, for each λ ∈˜ , the functional I λ admits at least two nontrivial critical points u λ,1 , u λ,2 ∈ X such that I λ (u λ,1 ) < 0 < I λ (u λ,2 ).

Main result
In this section we formulate and prove our main results concerning the existence of nontrivial bounded weak solutions to problem (1.1). First, we state the hypotheses on the nonlinearity f : × R → R. We suppose the following conditions: (H2) f : × R → R is a Carathéodory function satisfying the following conditions: (i) There exist ∈ (q, p * ) and constants κ 1 , κ 2 > 0 such that for a. a. x ∈ and for all s ∈ R; uniformly for a. a. x ∈ .
The energy functional I λ : W 1,H 0 ( ) → R of (1.1) is given by for all u ∈ W 1,H 0 ( ). It is clear that I λ ∈ C 1 and the critical points of I λ are the weak solutions of (1.1). Next, we introduce the functionals , : for all u ∈ W 1,H 0 ( ). We have that I λ = (u) − λ (u) and all these functionals are of class C 1 , where their derivatives are given by for all u, v ∈ W 1,H 0 ( ). First, we obtain the following proposition.  Adding (3.5) and (3.6) and recalling that p < q, we derive
Proof Let and be as given in (3.1). First we see that and fulfill all the required regularity properties in Theorem 2.4. Indeed, is coercive due to Proposition 2.1(iv) and the functional I λ is unbounded from below because of (H2)(ii). Also we see that It is clear that the interval is nonempty due to assumption (3.16). Hence, we can fix λ ∈ and we set where c p * is the best constant of the embedding W 1, p 0 ( ) → L p * ( ). Next, we define the functionũ where x 0 ∈ is such that B(x 0 , R) ⊆ , see (2.4). It is easy to see thatũ ∈ W 1,H 0 ( ).
Step 1 0 < (ũ) < r Using the representations of ω R in (2.5) and R in (2.6) we obtain From the definition of r and (3. 19) we see that we have to show that On the other hand, using (3.15), we get that (3.25) Combining (3.24), (3.16), (3.19) and (3.25) gives This proves Step 2. From Steps 1 and 2 and Proposition 3.3 we see that all the conditions in Theorem 2.4 are satisfied and so we conclude that problem (1.1) has at least two nontrivial weak Let us now consider the special case when f is nonnegative and independent of x. We suppose the following conditions: (H3) f : R → R is a continuous function with f (s) ≥ 0 for all s ∈ R satisfying the following conditions: (i) There exist ∈ (q, p * ) and constants κ 1 , κ 2 > 0 such that The next result is a consequence of Theorem 3.4.
Proof We are going to apply Theorem 3.4. First, we point out that, since f is nonnegative, we have F(t) ≥ 0 for all t ∈ R. So (3.15) is satisfied. In addition, we know that problem (1.1) has at least two nonnegative, nontrivial bounded weak solutions u λ , v λ ∈ W 1,H 0 ( ) such that I λ (u λ ) < 0 < I λ (v λ ).
Proof Let λ ∈ 2 be fixed. Then we can find ξ > 0 such that On the other hand condition (3.28) implies that lim sup t→0 + F(t) t p + t q = +∞.
Hence, we can find a number η ∈ (0, ξ) such that Then the assertion of the theorem follows from Corollary 3.5.
Finally, we want to give a concrete example for a function which fits in our setting. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.