An optimization problem for the first Steklov eigenvalue of a nonlinear problem

Abstract

In this paper we study the first (nonlinear) Steklov eigenvalue, $\lambda$, of the following problem: $-\Delta_{p}u + |u|^{p-2}u + \alpha\phi|u|^{p-2}u = 0$ in a bounded smooth domain $\Omega$ with $|\nabla u|^{p-2}\frac{\partial u}{\partial \nu} = \lambda |u|^{p-2}u$ on the boundary $ \partial\Omega$. We analyze the dependence of this first eigenvalue with respect to the weight $\phi$ and with respect to the parameter $\alpha$. We prove that for fixed $\alpha$ there exists an optimal $\phi_\alpha$ that minimizes $\lambda$ in the class of uniformly bounded measurable functions with fixed integral. Next, we study the limit of these minima as the parameter $\alpha$ goes to infinity and we find that the limit is the first Steklov eigenvalue in the domain with a hole where the eigenfunctions vanish.