Existence of Solution for Double-Phase Problem with Singular Weights

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 ðPλÞ 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:


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][2][3][4][5][6][7][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 ðP λ Þ 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 Ω is a smooth bounded domain in ℝ N , N ≥ 2, 0 < θ < 1, λ ∈ ℝ, a : Ω ↦ ½0, +∞Þ is Lipschitz continuous, and b 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][13][14]. In fact, Zhikov intended to provide models for strongly anisotropic materials in the context of homogenization. In particular, he considered the following functional: ð where the modulating coefficient aðxÞ dictates the geometry of the composite made of two differential materials, with hardening exponents p and q, 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][16][17][18][19][20][21], respectively, and references therein.
Example 1. The following function satisfies hypotheses Hð f Þ 1 : We are now in the position to state our main results. Firstly, problem ðP λ Þ has a solution when λ ≤ 0. Theorem 1. Assume that HðaÞ, HðbÞ, and Hð f Þ 1 hold. Then for all λ ≤ 0, problem ðP λ Þ has at least one nontrivial weak solution with negative energy.
Moreover, we also show that problem ðP λ Þ has a solution when λ > 0. In order to do this task, the following conditions are needed: Hð f Þ 2 : f : Ω × ℝ ⟶ ℝ is a Carathéodory function such that for a.a. x ∈ Ω, f ðx, 0Þ = 0, and (i) there exists C > 0 and d ∈ L s 2 ðΩÞ such that where 1 < r 2 < p < s 2 ; (ii) there exists a positive measurable subset Example 2. The following functions satisfy hypotheses Hð f Þ 2 : Theorem 2. Assume that HðaÞ, HðbÞ, and Hðf Þ 2 hold. Then for all λ ≥ 0, problem ðP λ Þ 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 W 1,H 0 ðΩÞ. In Section 3, the proof of the main results is given.

Preliminaries
In order to discuss problem ðPÞ, we need some facts on space W 1,H 0 ðΩÞ 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][23][24] and references therein.
The Musielak-Orlicz space L H ðΩÞ is defined by endowed with the Luxemburg norm juj H = inf fλ > 0 : By the above Proposition, there exists c τ > 0 such that juj τ ≤ c τ kuk, ∀u ∈ W H 0 ðΩÞ, where juj τ denotes the usual norm in L τ ðΩÞ for all 1 ≤ τ < Np/ðN − pÞ. It follows from (2) of Proposition 4 that j∇uj H is an equivalent norm in W 1,H 0 ðΩÞ. We will use the equivalent norm in the following discussion and write kuk = j∇uj H for simplicity.
In order to discuss the problem ðPÞ, we need to define a functional in W 1,H 0 ðΩÞ: We know that (see [25], P63, example) J ∈ C 1 ðW 1,H 0 ðΩÞ, ℝÞ and the double-phase operator is the derivative operator of J in the weak sense. Moreover, similar to the proof of Theorem 3.1 in [25], we know that the energy functional J is sequentially weakly lower semicontinuous.
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 λ ≤ 0, the functional φ λ is coercive on E.

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Hence, for any t ∈ ð0, t 0 Þ, from HðbÞ and Hð f Þ 1 (i), we deduce that Since The proof of Step 2 is now complete.
Let fu n λ g ⊂ E be a minimizing sequence of φ λ . Then, using Step 1, we get that fu n λ g is a bounded sequence. So, there exists u λ ∈ E such that, up to a subsequence, Recall that J is sequentially weakly lower semicontinuous, and so we deduce that Now, using Hölder's inequality, we get that, as n ⟶ +∞, Analogously, Hence, by (24) and (25), one yields Moreover, using assumptions Hð f Þ 1 (i) and Hð f Þ 1 (ii), for all ε > 0, there exists C ε > 0 such that The above information and Hölder's inequality imply Again, by Proposition 4 (1), we deduce that Thus, using the fact that fu n λ g is bounded in E and the dominated convergence theorem, we can infer that Advances in Mathematical Physics Hence, for every λ < 0, by (26) and (30), one yields which implies that φ λ 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 u λ is the required nontrivial solution of problem ðP λ Þ. Therefore, we complete the Proof of Theorem 1. Now, we are ready to prove Theorem 2.
The proof of Step 2 is now complete.
Let fu n λ g ⊂ E be a minimizing sequence of φ λ . Then, using Step 1, we get that fu n λ g is a bounded sequence. So, there exists u λ ∈ E such that, up to a subsequence, u n λ ⇀ u λ in E, u n λ ⟶ u λ in L p Ω ð Þ, u n λ x ð Þ ⇀ u λ x ð Þ a:e:in Ω:

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Thus, as the proof of Step 3 in Theorem 1, we also obtain that and φ λ 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 u λ is the required nontrivial solution of problem ðP λ Þ. 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.