A multiplicity result for quasilinear problems with convex and concave nonlinearities and nonlinear boundary conditions inunbounded domains

Kandilakis, Dimitrios A.

Electronic Journal of Differential Equations, Tome 2005 (2005).

Résumé

Summary: We study the following quasilinear problem with nonlinear boundary conditions $$\displaylines{ -\Delta_{p}u=\lambda a(x)|u|^{p-2}u+k(x)|u|^{q-2}u-h(x)|u|^{s-2}u, \quad \hbox{in }\Omega,\cr |\nabla u|^{p-2}\nabla u\cdot\eta+b(x)|u|^{p-2}u=0\quad \hbox{on }\partial\Omega, }$$ where $\Omega$ is an unbounded domain in $\mathbb{R}^{N}$ with a noncompact and smooth boundary $\partial\Omega, \eta$ denotes the unit outward normal vector on $\partial\Omega, \Delta_{p}u=\hbox{div\,}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian, $a, k, h$ and $b$ are nonnegative essentially bounded functions, $p^{\ast}$. The properties of the first eigenvalue $\lambda_{1}$ and the associated eigenvectors of the related eigenvalue problem are examined. Then it is shown that if $\lambda=\lambda_{1}$ it admits at least one nonnegative solution. Our approach is variational in character.

Détail

Classification : 35J20, 35J60
Keywords: variational method, fibering method, palais-Smale condition, genus