On a Class of Anisotropic Nonlinear Elliptic Equations with Variable Exponent

Since the variable exponent spaces have reflected into a various range of applications such as non-Newtonian fluids, plasticity, image processing, and nonlinear elasticity [1–4], some authors began to study the various properties of variable exponent space and some nonlinear problems with variable exponent growth. Edmunds et al. [5], Fan and Zhao [6] obtained that the variable exponent space Lp(x)(Ω) and


Introduction
Since the variable exponent spaces have reflected into a various range of applications such as non-Newtonian fluids, plasticity, image processing, and nonlinear elasticity [1][2][3][4], some authors began to study the various properties of variable exponent space and some nonlinear problems with variable exponent growth.Edmunds et al. [5], Fan and Zhao [6] obtained that the variable exponent space  () (Ω) and  ,() (Ω) are reflexive Banach spaces under suitable conditions on ().Later, Edmunds and Rákosník [7], Fan et al. [8] proved some continuous and compact Sobolev embedding theorems for the variable exponent spaces  ,() (Ω).For the anisotropic variable exponents spaces, in 2008, Mihȃilescu et al. [9] studied the eigenvalue problems for a class of anisotropic quasilinear elliptic equations with variable exponents.In 2011, Boureanu et al. [10] proved the existence of multiple solutions for a class of anisotropic elliptic equations with variable exponents.Recently, Stancu-Dumitru [11,12] has studied the existence of nontrivial solutions for a class of nonhomogeneous anisotropic problem.In particular, Fan [13] established some embedding theorems for anisotropic variable exponent Sobolev spaces.
The paper is organized as follows.In Section 2, we recall some results on anisotropic variable exponent Sobolev spaces and state our main results.In Section 3, we prove the existence, uniqueness and locally bounded of weak solution for the problem (1).In Section 4, the regularity of weak solutions for the problem (2) is proved.
Remark 4. Now, we give a simple example.

Now, we define the weak solution of the problem (1). A function
where and

The Proof of Theorem 6
In this section, we prove the regularity of weak solutions for the problem (2).