On singular double phase problem with variable exponents and convolution term

1 Introduction

In this paper, we investigate some existence results for the following singular double phase problem with variable exponents and convolution term of the form:

where the operator T p ( x ) , q ( x ) α ( x ) ( u ) : = div ∇ u p ( x ) − 2 ∇ u + α ( x ) ∇ u q ( x ) − 2 ∇ u is the double phase operator with variable exponents, Ω ⊂ R N is a bounded domain with smooth boundary ∂Ω, 0 ≤ α(⋅) ∈ L ^∞(Ω), λ is a positive real parameter. The function s ∈ C ( Ω ̄ ) is positive with compact support in Ω, μ : R N × R N → R is continuous function specified later and C may denote a positive constant and the same C may represent different constant.

Furthermore, throughout this paper, we denote with r + : = max x ∈ Ω ̄ r ( x ) and r − : = min x ∈ Ω ̄ r ( x ) for any function r ∈ C ( Ω ̄ ) , and we consider the following hypotheses:

1. 1 < p(x) < N and 0 < 1 − γ(x) ≪ p(x) ≪ q(x) ≪ h(x) ≪ p*(x) ≔ Np(x)/(N − p(x)) for a.e. x ∈ Ω, where p*(x) ≔ Np(x)/(N − p(x)) is the critical Sobolev exponent for variable exponents, the notation ℓ ₁(x) ≪ ℓ ₂(x) means that inf x ∈ Ω ( ℓ 2 ( x ) − ℓ 1 ( x ) ) > 0 . Moreover, we assume that

   q + 1 − γ + < 2 h − − q + 2 h + + γ − − 1 2 h − + γ + − 1 2 h + − p − .

2. μ : R N × R N → R is a continuous function such that

   1 < μ − ≤ μ + < N

   and there exists a function q ∈ C + R N such that

   1 p ( x ) + μ ( x , y ) N + 1 q ( y ) = 2 , ∀ x , y ∈ R N .

The study of this problem (1.1) is based on two reasons. On the one hand, double-phase problem has a wealthy physical background, on the grounds that the double phase operator has been utilized to describe steady-state solutions of reaction diffusion problems in biophysics, plasma physics, and chemical reaction analysis. The research on the double phase problems and applicable functionals originates from the seminal paper Zhikov [1]. Recently, Ambrosio and Rădulescu [2] had been concerned with a classification of fractional problems with uneven growth as follows:

where 0 < s < 1, ɛ > 0, ( − Δ ) t s is the fractional t-Laplacian operator, 2 ≤ p < q < N s and t ∈ {p, q}. For ɛ > 0 small enough, the authors showed multiple positive solutions and associated concentration properties related to the set of potential V reaching its minimum by suitable variational and topological arguments. In particular, the authors derived that there are at least five nontrivial smooth solutions to superlinear (p, q)-equations with indefinite potential and a concave boundary term, and they are all sign-informative and linearly ordered for small values of all parameters in [3].

In recent years, we note that the (p, q)-Laplacian equation has been extended to the (p(x), q(x))-Laplacian problem. For example, Alves et al. [4] studied the following equation:

where Ω is a bounded domain in R N ( N > 2 ) with C ² boundary ∂Ω and λ > 0 is a parameter. Here, Δ _p(x) and Δ _q(x) stand for the p(x)-Laplacian and q(x)-Laplacian operators, respectively. By using sub and supersolution methods, the authors derived the existence and regularity of solutions for the problem.

In addition, the variable exponential growth problem with singular nonlinear terms is widely used in non-Newtonian mechanics, cosmology, elasticity theory, electromagnetism and so on. This kind of problem has been widely studied in the past 30 years. In [5], Arora et al. proved the existence of two weak solutions for the singular double phase problem with variable exponents by the way of using the Nehari manifold technique primarily based on fibering maps. For some other interesting results can be found in [2],[6], [7], [8], [9], and the references therein.

On the other hand, the nonlinearity on the right-hand side of problem (1.1) is motivated by the equation

where I α : R N → R is a Riesz potential defined for x ∈ R N \ { 0 } by I α ( x ) = A α | x | N − α . Choquard delivered this equation to describe the electrons trapped in their own holes. Such equations also appear in the theory of stationary polarons in [10] and in the simulation of self-gravitating matter in [11]. Edmunds and Rákosník in [12] studied the existence and uniqueness of the minimizing solution of the problem

where f has subcritical growth. For critical case, Gao and Yang in [13],14] considered the following Brézis-Nirenberg type problem of the nonlinear Choquard equation:

where 2 μ * = 2 N − μ N − 2 , 0 < μ < N, the multiplicity results are obtained for this problem by using the variational method.

Alves et al. in [15] proved the Hardy-Littlewood-Sobolev inequality for variable exponents, and used it to study the quasilinear Choquard equation involving variable exponents. Zhang et al. in [16] obtained the existence of infinitely many solutions for the variable exponent critical reaction Choquard problem by using variational and analytical methods, including the variable exponent Hardy-Littlewood-Sobolev inequality and the concentration compactness principle for the variable growth problem. Some other results for the Choquard problem involving nonlocal operators can be founded in [8],9],[17], [18], [19], [20]. For the study of a class of partial differential equations with nonstandard growth, although many important results have been obtained at present, for Choquard equations involving (p(x), q(x))-Laplace operators and singular term, the results are still less, which is one of the motivations of this paper.

Inspired by the above literatures, this paper is devoted to the study of singular double phase problem with variable exponents and convolution term. The main difficulty is that due to the convolution term and the singular term, we adopt different technical means to study the existence of solutions to such problem. Inspired by [5], we use the Nehari manifold method based on the fibering mapping method to prove our existence results, which also supplement [5] to some extent.

Our main results are as follows:

The paper is organized as follows. First, Section 2 contains some definitions and properties of Musielak-Orlicz Sobolev spaces. Secondly, in Section 3 of this paper, we explore the properties of the fibering mapping corresponding to the energy functional, and prove some lemmas. Finally, in Section 4, the main theorem is proved by a series of lemmas.

2 Preliminaries

In this section, we introduce some symbols and related conclusions, which are very important to prove Theorem 1.1. Let C + ( Ω ̄ ) = { m ∈ C ( Ω ̄ ) : m ( x ) > 1 , ∀ x ∈ Ω ̄ } and S ( Ω ) represents the set of all measurable functions u : Ω → R . The variable Lebesgue space L ^m(⋅)(Ω), for m ∈ C ₊(Ω) is given by

and the norm is defined as

The space (L ^m(⋅)(Ω), ‖ ⋅‖ _m(⋅)) is separable and reflexive Banach space [17],21]. The conjugate space of L ^m(⋅)(Ω) is L ^m ′^(⋅)(Ω) with 1 m ( x ) + 1 m ′ ( x ) = 1 . For u ∈ L ^m(⋅)(Ω) and h ∈ L ^m ′^(⋅)(Ω), we have

Next, define M : Ω × [ 0 , ∞ ) by M ( x , s ) : = s p ( x ) + λ ( x ) s q ( x ) for all (x, s) ∈ Ω × [0, ∞). The Musielak space L M ( Ω ) is given by

L M ( Ω ) is equipped with the norm

The Musielak space L M ( Ω ) is a reflexive Banach space [10]. The Musielak-Orlicz Sobolev space W 1 , M ( Ω ) is defined as

W 1 , M ( Ω ) is equipped with the norm

and W 0 1 , M ( Ω ) is defined as the completion of C c ∞ ( Ω ) in W 1 , M ( Ω ) . For u ∈ W 0 1 , M ( Ω ) , if (H ₁) holds, then we have ‖ u ‖ H ≤ C ∇ u M , for some constant C > 0 [10]. The space W 0 1 , M ( Ω ) can be equipped with the norm  ‖ u ‖ = ∇ u M . W 0 1 , M ( Ω ) and W 1 , M ( Ω ) are separable reflexive Banach spaces.

Next, inspired by [22], Theorem 2.3, 2.4], we have the following results:

Then we give the Hardy-Littlewood-Sobolev inequality for variable exponents, see [15].

3 Some useful lemmas

The energy functional J ( u ) : W 0 1 , H ( Ω ) → R corresponding to problem (1.1) is given by

If u ∈ W 0 1 , H ( Ω ) is a weak solution of problem (1.1), it means that there hold s(x)|u|^−γ(x) φ ∈ L ¹(Ω) and

for all φ ∈ W 0 1 , H ( Ω ) . The functional J ( u ) is not C ¹ because of the singular term s ( x ) | u | 1 − γ ( x ) 1 − γ ( x ) . Hence, the fibering map ϕ u : [ 0 , ∞ ) → R we defined by ϕ u ( t ) = J ( t u ) , ∀t ≥ 0. We have ϕ _u ∈ C ^∞(0, ∞) and

To solve problem (1.1), the Nehari manifold N λ is defined as

Next, we split N _λ into three constrained subsets

Next, we discuss the properties of fibering maps in all possible cases in [23]. Define

We have

and

For β > ζ > δ > 0 and λ, m, n, r > 0, choose the function l _λ (t):= t ^ζ m − λt ^−δ n − t ^β r. For t > 0, l _λ (t) = 0 ⇔ t ^δ l _λ (t) = 0, i.e., t ^ζ+δ m − t ^β+δ r = λn. Let f(t) = t ^ζ+δ m − t ^β+δ r and g(t) = λn. The function f(t) satisfies

If we choose λ > 0 and sufficiently small then l _λ (t) has exactly two solutions t ₁, t ₂ with t ₁ < t ₂, such that l λ ′ ( t 1 ) > 0 and l λ ′ ( t 2 ) < 0 .

In view of the inequalities (3.3) and (3.4), we note that the graph of ϕ u ′ ( t ) lies between the graphs of ν λ , u = t q + − 1 m u − λ t − γ + n u − t 2 h − − 1 r u and ω λ , u ( t ) = t p − − 1 m u − λ t − γ − n u − t 2 h + − 1 r u . Therefore, there exists at least two critical points of ϕ _u (t), namely, t ₁ < t ₂, such that ϕ _u (t) has local minima at t ₁ = t ₁(u) and at local maxima at t ₂ = t ₂(u). If u ∈ N λ + , t ₁ = 1 < t ₂, t 2 u ∈ N λ − and for u ∈ N λ − , t ₁ < t ₂ = 1, such that t 1 u ∈ N λ + . Also, ϕ _u is decreasing in (0, t ₁), increasing on (t ₁, t ₂), and again decreasing on (t ₂, ∞). Thus, there exists λ _* > 0, such that for 0 < λ < λ _*, we have ϕ u ′ ( t ) = 0 , i.e., t u ∈ N λ but t u ∉ N λ 0 .