Poincare Inequalities and Neumann Problems for the Variable Exponent Setting

We extend the results of [5], where we proved an equivalence between weighted Poincare inequalities and the existence of weak solutions to a family of Neumann problems related to a degenerate $p$-Laplacian. Here we prove a similar equivalence between Poincare inequalities in variable exponent spaces and solutions to a degenerate $p(x)$-Laplacian, a non-linear elliptic equation with nonstandard growth conditions.