Characterization of the variable exponent Sobolev norm without derivatives

The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. Since the difference quotient is based on shifting the function, it cannot be generalized to the variable exponent case. In its place, we introduce a smoothed difference quotient and show that it can be used to characterize the variable exponent Sobolev space.

The Existence of Solutions to the NonhomogeneousA-Harmonic Equations with Variable Exponent

We first discuss the existence and uniqueness of weak solution for the obstacle problem of the nonhomogeneousA-harmonic equation with variable exponent, and then we obtain the existence of the solutions of the equationd⋆A(x,dω)=B(x,dω)in the weighted variable exponent Sobolev spaceWdp(x)(Ω,Λl,μ).

Three solutions to a p(x)-Laplacian problem in weighted-variable-exponent Sobolev space

In this paper, we verify that a general p(x)-Laplacian Neumann problem has at least three weak solutions, which generalizes the corresponding result of the reference [R. A. Mashiyev, Three Solutions to a Neumann Problem for Elliptic Equations with Variable Exponent, Arab. J. Sci. Eng. 36 (2011) 1559-1567].