Stability of principal eigenvalue of the Schrödinger type problem for differential inclusions

Abstract

Let $\Omega\subset \mathbb R^3$ be a bounded domain. Denote by $\lambda_1(m)$ the principal eigenvalue of the Schrödinger operator $L_m(u)=-\nabla^2 u-mu$ defined on $H^1_0(\Omega)\cap W^{2,1}(\Omega)$. We prove that $\lambda_1: L^{3/2}(\Omega)\to \mathbb R$ is continuous.