Quasilinear equations involving nonlinear Neumann boundary conditions
2012; Khayyam Publishing; Volume: 25; Issue: 7/8 Linguagem: Inglês
10.57262/die/1356012656
ISSN0893-4983
AutoresLeonelo Iturriaga, Sebastián Lorca, Eugenio Saavedra, Pedro Ubilla,
Tópico(s)Nonlinear Differential Equations Analysis
ResumoWe 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.
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