Infinitely many weak solutions for a p-Laplacian equation with nonlinear boundary conditions

Abstract

We study the following quasilinear problem with nonlinear boundary conditions $$displaylines -Delta _{p}u+a(x)|u|^{p-2} u=f(x,u) quad mbox{in }Omega, cr | abla u|^{p-2} frac{partial u}{partial u}=g(x,u) quad mbox{on } partialOmega, }$$ where $Omega$ is a bounded domain in $mathbb{R}^{N}$ with smooth boundary and $frac{partial}{partial u}$ is the outer normal derivative, $Delta_{p}u=mathop{m div}(| abla u|^{p-2} abla u)$ is the $p$-Laplacian with $1<p<N$. We consider the above problem under several conditions on $f$ and $g$, where $f$ and $g$ are both Carath'{e}odory functions. If $f$ and $g$ are both superlinear and subcritical with respect to $u$, then we prove the existence of infinitely many solutions of this problem by using ``fountain theorem'' and ``dual fountain theorem'' respectively. In the case, where $g$ is superlinear but subcritical and $f$ is critical with a subcritical perturbation, namely $f(x,u)=|u|^{p^{*}-2}u+lambda|u|^{r-2}u$, we show that there exists at least a nontrivial solution when $p<r<p^{*}$ and there exist infinitely many solutions when $1<r<p$, by using ``mountain pass theorem'' and ``concentration-compactness principle'' respectively.

Keywords

• p-Laplacian
• nonlinear boundary conditions
• weak solutions
• critical exponent
• variational principle