Compactness results for quasilinear problems with variable exponent on the whole space

Abstract

In this work we give a compactness result which allows us to prove the point-wise convergence of the gradients of a sequence of solutions to a quasilinear inequality and for an arbitrary open set. This result suggests solutions to many problems, notably nonlinear elliptic problems with critical exponent.

Keywords

• p(x)-Laplace operator
• critical Sobolev exponent
• compactness
• Palais-Smale condition