Electron. J. Diff. Eqns., Vol. 1998(1998), No. 10, pp. 1-13.

Existence and multiplicity of solutions to a p-Laplacian equation with nonlinear boundary condition

Klaus Pflueger

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.

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Klaus Pflueger
FB Mathematik, Freie Universitat Berlin
Arnimallee 3, 14195 Berlin, Germany
e-mail: pflueger@math.fu-berlin.de
Web page: http://www.math.fu-berlin.de/user/pflueger

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