Principal eigenvalues of elliptic problems with singular potential and bounded weight function

Abstract

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ with $C^{0,1}$ boundary, and let $d_{\Omega}:\Omega\rightarrow\mathbb{R}$ be the distance function $d_{\Omega}\left( x\right) :=dist\left( x,\partial\Omega\right) .$ Our aim in this paper is to study the existence and properties of principal eigenvalues of self-adjoint elliptic operators with weight function and singular potential, whose model problem is $-\Delta u+bu=\lambda mu$ in $\Omega,$ $u=0$ on $\partial\Omega,$ $u>0$ in $\Omega,$ where $b:\Omega \rightarrow\mathbb{R}$ is a nonnegative function such that $d_{\Omega}^{2}b\in L^{\infty}\left( \Omega\right) ,$ $m:\Omega\rightarrow\mathbb{R}$ is a nonidentically zero function in $L^{\infty}\left( \Omega\right) $ that may change sign, and the solutions are understood in weak sense.