Removability of singularity for nonlinear elliptic equations with p(x)-growth ∗

Using Moser’s iteration method, we investigate the problem of removable isolated singularities for elliptic equations with p(x)-type nonstandard growth. We give a sucient condition for removability of singularity for the equations in the framework of variable exponent Sobolev spaces.


Introduction
In recent years, the research of elliptic equations with variable exponent growth conditions has been an interesting topic. These problems possess very complicated nonlinearities, for instance, the p(x)-Laplacian operator −div(|∇u| p(x)−2 ∇u) is inhomogeneous, and these problems have many important applications, see [1,2,3]. Since Kováčik and Rákosník first studied the L p(x) spaces and W k,p(x) spaces in [4], many results have been obtained concerning these kinds of variable exponent spaces, see examples in [5 − 12].
Recently, there have been a few papers on the study of the removability of singularities for the equations with nonstandard growth. Lukkari [20] investigated the removability of a compact set for the equation −div |Du| p(x)−2 Du = 0. For the anisotropic elliptic equation, the removability of a compact set was proved by Cianci [21]. Cataldo and Cianci [22] considered the conditions of removability of an isolated singular point for equation (1.1) in the case of g(x, u) = |u| q−2 u.
In this paper, following Moser's method [23], we establish the condition to ensure the removability of singularities.

Preliminaries
We first recall some facts on spaces L p(x) and W k,p(x) . For the details see [4,8].
Let P(Ω) be the set of all Lebesgue measurable functions p : Ω → [1, ∞], we denote where Ω ∞ = {x ∈ Ω : p(x) = ∞}. The variable exponent Lebesgue space L p(x) (Ω) is the class of all functions u such that ρ p(x) (tu) < ∞, for some t > 0. L p(x) (Ω) is a Banach space equipped with the norm For any p ∈ P(Ω), we define the conjugate function p (x) as Theorem 2.2 Let p ∈ P(Ω) with p + < ∞. For any u ∈ L p(x) (Ω), we have The variable exponent Sobolev space W 1,p(x) (Ω) is the class of all functions u ∈ L p(x) (Ω) such that |∇u| ∈ L p(x) (Ω). W 1,p(x) (Ω) is a Banach space equipped with the norm We say that the function u(x) belongs to the space W From Zhikov [5,6], we know smooth functions are not dense in W 1,p(x) (Ω) without additional assumptions on the exponent p(x). To study the Lavrentiev phenomenon, he considered the following log-Hölder continuous condition for all x, y ∈ Ω such that |x − y| ≤ 1 2 . If the log-Hölder continuous condition holds, then smooth functions are dense in W 1,p(x) (Ω) and we can define the Sobolev spaces with zero boundary values W 1,p(x) 0 (Ω), as the closure of C ∞ 0 (Ω) with the norm of · W 1,p(x) (Ω) .
Proof. As q(x), p(x) are continuous on Ω, for ε 1 ∈ (0, 1) and 0 ∈ Ω, there exists δ > 0 such that |q(0) − q(y)| < ε 1 and |p(0) − p(y)| < ε 1 whenever |y| < δ. For any y ∈ B δ (0) ∩ Ω, we have we know the set S is bounded above. By the Continuum Property, it has a smallest upper bound δ 0 . This smallest upper bound δ 0 is called the supremum of the set S. We write in Ω \ {0}, the following equality is true: We say that the solution u(x) of equation (1.1) has a removable singularity at the point 0

Proof of theorems
In this section we state and prove the following theorems. In the sequel by C we denote a constant, the value of which may vary from line to line.
Proof. For ρ < R we define a smooth cut-off function ϕ 1 (x) satisfying conditions: m, n ≥ 0 are nonnegative numbers to be determined later, and then We substitute the test function ψ(x) into the integral identity (2.3), we obtain By virtue of the conditions (1.2) − (1.5),