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July/August 2012 Quasilinear equations involving nonlinear Neumann boundary conditions
Leonelo Iturriaga, Sebastián Lorca, Eugenio Saavedra, Pedro Ubilla
Differential Integral Equations 25(7/8): 657-664 (July/August 2012).

Abstract

We study the multiplicity of positive solutions of the problem $$ -\Delta_p u+|u|^{p-2}u=0 $$ in a bounded smooth domain $\Omega\subset{\mathbb{R}}^N$, with a nonlinear boundary condition given by $$ |\nabla u|^{p-2}\partial u/\partial\nu=\lambda f(u) +\mu\varphi(x)|u|^{q-1}u, $$ where $f$ is continuous and satisfies some kind of $p-$superlinear condition at 0 and $p-$sublinear condition at infinity, $0<q< p-1$ and $\varphi$ is $L^\beta(\partial\Omega)$ for some $\beta>1$. In addition, we consider the case $q=0$, where the nonlinear boundary condition becomes an elliptic inclusion. Our approach allows us to show that these problems have at least six nontrivial solutions, three positive and three negative, for some positive parameters $\lambda$ and $\mu$. The proof is based on variational arguments.

Citation

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Leonelo Iturriaga. Sebastián Lorca. Eugenio Saavedra. Pedro Ubilla. "Quasilinear equations involving nonlinear Neumann boundary conditions." Differential Integral Equations 25 (7/8) 657 - 664, July/August 2012.

Information

Published: July/August 2012
First available in Project Euclid: 20 December 2012

zbMATH: 1265.35075
MathSciNet: MR2975688

Subjects:
Primary: 35J25, 35J60, 35J70

Rights: Copyright © 2012 Khayyam Publishing, Inc.

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Vol.25 • No. 7/8 • July/August 2012
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