An Approximation of Hedberg’s Type in Sobolev Spaces with Variable Exponent and Application

Abstract

The aim of this paper is to extend the usual framework of PDE with Au=-div ax,u,∇u to include a large class of cases with Au=∑β≤α-1βDβAβx,u,∇u,…,∇αu, whose coefficient Aβ satisfies conditions (including growth conditions) which guarantee the solvability of the problem Au=f. This new framework is conceptually more involved than the classical one includes many more fundamental examples. Thus our main result can be applied to various types of PDEs such as reaction-diffusion equations, Burgers type equation, Navier-Stokes equation, and p-Laplace equation.