Electronic Journal of Differential Equations,
Vol. 1998(1998), No. 10, pp. 1-13.
Title: Existence and multiplicity of solutions to a p-Laplacian
equation with nonlinear boundary condition
Author: Klaus Pflueger (Freie Univ. Berlin, Germany)
Abstract:
We study the nonlinear elliptic boundary value problem
$$ A u = f(x,u) \quad {\rm in }\Omega\,,$$
$$ Bu = g(x,u) \quad {\rm on }\partial \Omega\,,$$
where $A$ is an operator of p-Laplacian type, $\Omega$ is
an unbounded domain in ${\Bbb R}^N$ with non-compact boundary, and
$f$ and $g$ are subcritical nonlinearities.
We show existence of a nontrivial nonnegative weak solution
when both $f$ and $g$ are superlinear. Also we show
existence of at least two nonnegative solutions when one of the two
functions $f$, $g$ is sublinear and the other one superlinear.
The proofs are based on variational methods applied to weighted
function spaces.
Submitted March 5, 1998. Published April 10, 1998.
Math Subject Classification: 35J65, 35J20.
Key Words: p-Laplacian; nonlinear boundary condition; variational methods;
unbounded domain; weighted function space.