Isolation and simplicity for the first eigenvalue of the \(p\)-Laplacian with a nonlinear boundary condition (Q1608806)

Summary: We prove the simplicity and isolation of the first eigenvalue for the problem \(\Delta_p u= | u|^{p-2}u\) in a bounded smooth domain \(\Omega \supset {\mathbb R}^{N}\), with a nonlinear boundary condition given by \(| \nabla u|^{p-2}\partial u/\partial v=\lambda | u|^{p-2}u\) on the boundary of the domain.