Bifurcation from the first eigenvalue of the p-Laplacian with nonlinear boundary condition

Abstract

We consider the problem $$\displaylines{ \Delta_{p}u =|u|^{p-2}u \quad\text{in }\Omega, \cr |\nabla u|^{p-2}\frac{\partial u}{\partial \nu}=\lambda|u|^{p-2}u + g(\lambda,x,u) \quad\text{on }\partial\Omega, }$$ where $\Omega$ is a bounded domain of $\mathbb{R}^{N}$ with smooth boundary, $N\geq 2$, and $\Delta_p$ denotes the p-Laplacian operator. We give sufficient conditions for the existence of continua of solutions bifurcating from both zero and infinity at the principal eigenvalue of p-Laplacian with nonlinear boundary conditions. We also prove that those continua split on two, one containing strictly positive and the other containing strictly negative solutions. As an application we deduce results on anti-maximum and maximum principles for the p-Laplacian operator with nonlinear boundary conditions.

Keywords

• Bifurcation theory
• topological degree
• $p$-Laplacian
• elliptic problem
• nonlinear boundary condition
• maximum and anti-maximum principles