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

Zhao, Ji-Hong ; Zhao, Pei-Hao

Electronic Journal of Differential Equations, Tome 2007 (2007).

Résumé

Summary: 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 1consider 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 $p$ and there exist infinitely many solutions when 1, by using "mountain pass theorem" and "concentration-compactness principle" respectively.

Détail

Classification : 35J20, 35J25
Keywords: p-Laplacian, nonlinear boundary conditions, weak solutions, critical exponent, variational principle