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, $q<p<s$ 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
 $\lambda<\lambda_{1}$, 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.