Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 57, pp. 1-12.
Title: A multiplicity result for quasilinear problems with
convex and concave nonlinearities and nonlinear
boundary conditions in unbounded domains
Author: Dimitrios A. Kandilakis (Technical Univ. Of Crete, Greece)
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 $\partial\Omega$,
$\Delta_{p}u=\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$ is the
$p$-Laplacian, $a$, $k$, $h$ and $b$ are nonnegative essentially
bounded functions, $q