Electron. J. Diff. Eqns., Vol. 2005(2005), No. 57, pp. 1-12.

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

Dimitrios A. Kandilakis

Abstract:
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, $q$ less than $p$ less than $s$ and $p^{\ast}<s$. 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 $\lambda$ less than $\lambda_{1}$, the original problem admits an infinite number of solutions one of which is nonnegative, while if $\lambda=\lambda_{1}$ it admits at least one nonnegative solution. Our approach is variational in character.

Submitted September 27, 2004. Published May 31, 2005.
Math Subject Classifications: 35J20, 35J60.
Key Words: Variational method; fibering method; Palais-Smale condition; genus.

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Dimitrios A. Kandilakis
Department of Sciences
Technical University Of Crete
Chania, Crete, 73100 Greece
email: dkan@science.tuc.gr

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