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

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