Differential and Integral Equations

Quasilinear equations involving nonlinear Neumann boundary conditions

Leonelo Iturriaga, Sebastián Lorca, Eugenio Saavedra, and Pedro Ubilla

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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.

Article information

Differential Integral Equations, Volume 25, Number 7/8 (2012), 657-664.

First available in Project Euclid: 20 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations 35J25: Boundary value problems for second-order elliptic equations 35J70: Degenerate elliptic equations


Iturriaga, Leonelo; Lorca, Sebastián; Saavedra, Eugenio; Ubilla, Pedro. Quasilinear equations involving nonlinear Neumann boundary conditions. Differential Integral Equations 25 (2012), no. 7/8, 657--664. https://projecteuclid.org/euclid.die/1356012656

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