Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 90, pp. 1-14.

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

Authors: Ji-Hong Zhao (Lanzhou Univ., Lanzhou, China)
         Pei-Hao Zhao (Lanzhou Univ., Lanzhou, China)

Abstract:
 We study the following quasilinear problem with
 nonlinear boundary conditions
 $$\displaylines
 -\Delta _{p}u+a(x)|u|^{p-2} u=f(x,u) \quad \mbox{in }\Omega, \cr
 |\nabla u|^{p-2} \frac{\partial u}{\partial \nu}=g(x,u) \quad
 \mbox{on } \partial\Omega,
 }$$
 where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with smooth
 boundary and  $\frac{\partial}{\partial \nu}$ is the outer normal
 derivative, $\Delta_{p}u=\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$ 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.
  
Submitted March 26, 2007. Published June 15, 2007.
Math Subject Classifications: 35J20, 35J25.
Key Words: p-Laplacian; nonlinear boundary conditions; weak solutions; 
           critical exponent; variational principle.