Nonexistence results for differential inequalities involving $A$-Laplacian

Abstract

We give a sufficient condition for nonexistence of nontrivial nonnegative solutions to the partial differential inequality involving $A$-Laplacian: $-\Delta_A u\ge \Phi (u)$, where $u$ is defined on ${{\mathbb{R}^{n}}}$. The condition obtained relies on the rate of decay at infinity of certain functions involving $A$ and $\Phi$. The techniques, based on methods due to Mitidieri and Pohozaev, exploit suitable a priori estimates in the framework of Orlicz-Sobolev spaces. The result is illustrated by logarithmic $A$-Laplacians and logarithmic functions $\Phi$.