On a class of nonlinear elliptic problems with Neumann boundary conditions growing like a power

Chipot, M; Voirol, F (1995). On a class of nonlinear elliptic problems with Neumann boundary conditions growing like a power. Zeitschrift für Analysis und ihre Anwendungen, 14(4):853-868.

Abstract

We investigate the issue of existence and the number of solutions for the problem $\Delta u=au^p$ in $\Omega,\ u=0$ on $\Gamma_0,\ \partial u/\partial n=u^q$ on $\Gamma_1$, where $\Gamma_0$ and $\Gamma_1$ are two parts of the boundary of the open set $\Omega$. In dimension one we are able to find all the solutions to the problem. In higher dimensions we give existence and non-existence results for different solutions depending on $p,q$ and $\Omega$.

We investigate the issue of existence and the number of solutions for the problem $\Delta u=au^p$ in $\Omega,\ u=0$ on $\Gamma_0,\ \partial u/\partial n=u^q$ on $\Gamma_1$, where $\Gamma_0$ and $\Gamma_1$ are two parts of the boundary of the open set $\Omega$. In dimension one we are able to find all the solutions to the problem. In higher dimensions we give existence and non-existence results for different solutions depending on $p,q$ and $\Omega$.