Existence and multiplicity of solutions to a p-Laplacian equation with nonlinear boundary condition

Abstract

We study the nonlinear elliptic boundary value problem $$ A u = f(x,u) quad { m in }Omega,,$$ $$ Bu = g(x,u) quad { m on }partial Omega,,$$ where $A$ is an operator of p-Laplacian type, $Omega$ is an unbounded domain in ${Bbb R}^N$ with non-compact boundary, and $f$ and $g$ are subcritical nonlinearities. We show existence of a nontrivial nonnegative weak solution when both $f$ and $g$ are superlinear. Also we show existence of at least two nonnegative solutions when one of the two functions $f$, $g$ is sublinear and the other one superlinear. The proofs are based on variational methods applied to weighted function spaces.

Keywords

• p-Laplacian
• nonlinear boundary condition
• variational methods
• unbounded domain
• weighted function space.