Electron. J. Diff. Eqns., Vol. 2007(2007), No. 90, pp. 1-14.

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

Ji-Hong Zhao, Pei-Hao Zhao

Abstract:
We study the following quasilinear problem with nonlinear boundary conditions
$$\displaylines{ -\Delta _{p}u+a(x)|u|^{p-2} u=f(x,u) \quad \hbox{in }\Omega, \cr |\nabla u|^{p-2} \frac{\partial u}{\partial \nu}=g(x,u) \quad \hbox{on } \partial\Omega, }$$
where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with smooth boundary and $\frac{\partial}{\partial \nu}$ is the outer normal derivative, $\Delta_{p}u=\hbox{\rm div}(|\nabla u|^{p-2}\nabla u)$ 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 Caratheodory 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.

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Ji-Hong Zhao
Department of Mathematics
Lanzhou University
Lanzhou, 730000, China
email: zhaojihong2007@yahoo.com.cn
Pei-Hao Zhao
Department of Mathematics
Lanzhou University
Lanzhou, 730000, China
email: zhaoph@lzu.edu.cn

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