Existence Results in Weighted Sobolev Spaces for Some Singular Quasilinear Elliptic Equations

Abstract

In this paper, we obtain the existence of a nontrivial solution for a class of singular quasilinear elliptic equations in weighted Sobolev spaces. The proofs rely on Galerkin-type techniques, Brouwer fixed point theorem, and a new weighted compact Sobolev-type embedding theorem established by Shapiro. The equation is one of the most useful sets of Navier-Stokes equations, which describe the motion of viscous fluid substances such as liquids, gases and so on.

Existence of Solutions for a Class of Quasilinear Parabolic Equations with Superlinear Nonlinearities

“…In [4], Jia and Zhao obtained the existence of a nontrivial solution for a class of singular quasilinear elliptic equations in weighted Sobolev spaces.…”

“…There are a number of results concerning solvability of different boundary problems for quasilinear equations (elliptic and parabolic) in which the nonlinearities satisfy sublinear or linear conditions in the weighted Sobolev space, for example, [1][2][3][4][5][6].…”