Existence of Solution for Double-Phase Problem with Singular Weights

Abstract

The aim of this paper is to establish the existence of solutions for singular double-phase problems depending on one parameter. This work improves and complements the existing ones in the literature. There seems to be no results on the existence of solutions for singular double-phase problems.

1. Introduction and Main Results

The study of various mathematical problems involving the double-phase operator has become very attractive in recent decades. The existence and multiplicity of solutions of double-phase Dirichlet problems has been studied by several authors (see, e.g., [1–8]); in particular, for the eigenvalues of the double-phase operator, see [7]. For other double-phase problems with variable exponents, there are the works of Zhang and Radulescu [9], Shi et al. [10], and Cencelj et al. [11].

But up to now, to the best of our knowledge, no paper discussing the existence of solutions for singular double-phase problems via critical point theory can be found in the existing literature. In order to fill in this gap, we study double-phase problems from a more extensive viewpoint. More precisely, we are going to prove that problem has at least one solution. To the best of our knowledge, this is one of the first works which combines a singular term and indefinite term in one problem.

This paper is concerned with the existence of solutions to the following singular double-phase problem: where is a smooth bounded domain in , , , , is Lipschitz continuous, and is a given measurable function. The precise conditions on the data will be presented later.

Problems of the above type arise for instance in nonlinear elasticity. The main reasons are to describe the behavior of Lavrentiev’s phenomenon; we refer to [12–14]. In fact, Zhikov intended to provide models for strongly anisotropic materials in the context of homogenization. In particular, he considered the following functional: where the modulating coefficient dictates the geometry of the composite made of two differential materials, with hardening exponents and , respectively. Recently, there is a wide literature on the regularity theory for minimizers of variational problems and solutions of differential equations with the double-phase operator; far from being complete, we refer the readers to [15–21], respectively, and references therein.

In the entire paper, we suppose the following assumptions:

: is Lipschitz continuous and are chosen such that .

: such that in .

: is a continuous function such that for a.a. , , and (i)there exists positive measurable subsets and such that on , and where ;(ii)there exists such that on , and where ;(iii)there exists such that where .

Example 1. The following function satisfies hypotheses : with .

We are now in the position to state our main results. Firstly, problem has a solution when .

Theorem 1. Assume that , , and hold. Then for all , problem has at least one nontrivial weak solution with negative energy.

Moreover, we also show that problem has a solution when . In order to do this task, the following conditions are needed:

: is a Carathéodory function such that for a.a. , , and (i)there exists and such that where ;(ii)there exists a positive measurable subset such that

Example 2. The following functions satisfy hypotheses :

Theorem 2. Assume that , , and hold. Then for all , problem has at least one nontrivial weak solution with negative energy.

The rest of this paper is organized as follows. In Section 2, we present some necessary preliminary knowledge on space . In Section 3, the proof of the main results is given.

2. Preliminaries

In order to discuss problem , we need some facts on space which are called Musielak-Orlicz-Sobolev spaces. For this reason, we will recall some properties involving the Musielak-Orlicz spaces, which can be found in [7, 22–24] and references therein.

Denote by the set of all generalized -function. For and , we define It is clear that is a locally integrable function and which is called condition .

The Musielak-Orlicz space is defined by endowed with the Luxemburg norm The Musielak-Orlicz-Sobolev space is defined by and it is equipped with the norm We denote by the completion of in . With these norms, the spaces , , and are separable reflexive Banach spaces (see [7] for the details).

Proposition 3. ([1], Proposition 2.1).
Set For , we have (i)For (ii)(iii)If , then (iv)If , then

Proposition 4. ([7], Proposition 2.15, Proposition 2.18). (1)If , then the embedding from to is continuous. In particular, if , then the embedding is compact(2)Assume that holds. Then, Poincare’s inequality holds; that is, there exists a positive constant such that

By the above Proposition, there exists such that , where denotes the usual norm in for all . It follows from (2) of Proposition 4 that is an equivalent norm in . We will use the equivalent norm in the following discussion and write for simplicity.

In order to discuss the problem , we need to define a functional in :

We know that (see [25], P63, example) and the double-phase operator is the derivative operator of in the weak sense. Moreover, similar to the proof of Theorem 3.1 in [25], we know that the energy functional is sequentially weakly lower semicontinuous.

3. Variational Setting and Proof of the Main Results

For any and each , we define where . By using , we get and . Also, by Proposition 4 (1) we deduce that embeddings and are compact and continuous. Furthermore, there exists a constant such that

Now, we are ready to prove Theorem 1.

Proof of Theorem 1. To complete the proof of the main result, we need to consider the following three steps.

Step 1. We first show that for every , the functional is coercive on .
Let be fixed. Put . Clearly, from the continuity of , there exists such that Thus, we deduce that for any and , By virtue of assumption (iii), (15), (17), and Proposition 3, one has for any with Since and , so this implies as . The proof of Step 1 is now completed.

Step 2. We show that there exists with , for small enough.
Let such that , in a subset , and in . Thus, by condition (i), it follows that there exists such that Hence, for any , from and (i), we deduce that Since , we have for with The proof of Step 2 is now complete.

Step 3. We show that there exists such that for any .
Let be a minimizing sequence of . Then, using Step 1, we get that is a bounded sequence. So, there exists such that, up to a subsequence, Recall that is sequentially weakly lower semicontinuous, and so we deduce that Now, using Hölder’s inequality, we get that, as , Analogously, Hence, by (24) and (25), one yields Moreover, using assumptions (i) and (ii), for all , there exists such that The above information and Hölder’s inequality imply Again, by Proposition 4 (1), we deduce that Thus, using the fact that is bounded in and the dominated convergence theorem, we can infer that Hence, for every , by (26) and (30), one yields which implies that is weakly lower semicontinuous, and consequently, which implies that So, we complete Step 3.

Therefore, combining the above Steps 2 and 3, we deduce that is the required nontrivial solution of problem . Therefore, we complete the Proof of Theorem 1.

Now, we are ready to prove Theorem 2.

Proof of Theorem 2. To complete the proof of the main result, we need to consider the following three steps.

Step 1. We first show that for every , the functional is coercive on .
Firstly, due to condition (i), one has Again, using the condition (i), Hölder’s inequality, Proposition 3, and relation (34), we deduce that for any with , the following inequality holds true: Since , we infer that as . The proof of Step 1 is now complete.

Step 2. We show that there exists with , for small enough.
Let such that , in a subset , and in . Thus, by condition (ii), it follows that Hence, for any , from and (ii), we deduce that Since , we have for with The proof of Step 2 is now complete.

Step 3. We show that there exists such that for any .
Let be a minimizing sequence of . Then, using Step 1, we get that is a bounded sequence. So, there exists such that, up to a subsequence, Thus, as the proof of Step 3 in Theorem 1, we also obtain that and is weakly lower semicontinuous, and consequently, which implies that The proof of Step 3 is complete.

Therefore, combining the above Steps 2 and 3, we deduce that is the required nontrivial solution of problem . Thus, we complete the Proof of Theorem 2.

Data Availability

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.

Acknowledgments

This work is supported by the National Key Research and Development Program of China (No. 2018YFC0310500), the Fundamental Research Funds for the Central Universities (No. 3072020CF2401), the Natural Science Foundation of Inner Mongolia (No. 2017MS0116), the National Natural Science Foundation of China (No. 11201095), the Postdoctoral Research Startup Foundation of Heilongjiang (No. LBH-Q14044), and the Science Research Funds for Overseas Returned Chinese Scholars of Heilongjiang Province (No. LC201502).