Existence problems for the $p$-Laplacian

Abstract

We consider a number of boundary value problems involving the $p$-Laplacian. The model case is $-{\Delta }_{p}u=V|u{|}^{p-2}u$ for $u\in W_0^{1,2}(D)$ with $D$ a bounded domain in ${\mathbf{R}}^n$. We derive necessary conditions for the existence of nontrivial solutions. These conditions usually involve a lower bound for a product of powers of the norm of $V$, the measure of $D$, and a sharp Sobolev constant. In most cases, these inequalities are best possible. Applications to non-linear eigenvalue problems are also discussed.