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

Authors : Tomas GODOY

Pages : 107-127

Doi:10.33205/cma.1272110

View : 29 | Download : 37

Publication Date : 2023-06-15

Article Type : Research Paper

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 $d_{\\Omega}\\leftinsert ignore into journalissuearticles values( x\\right); :=dist\\leftinsert ignore into journalissuearticles values( x,\\partial\\Omega\\right); .$ 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 $d_{\\Omega}^{2}b\\in L^{\\infty}\\leftinsert ignore into journalissuearticles values( \\Omega\\right); ,$ $m:\\Omega\\rightarrow\\mathbb{R}$ is a nonidentically zero function in $L^{\\infty}\\leftinsert ignore into journalissuearticles values( \\Omega\\right); $ that may change sign, and the solutions are understood in weak sense.
Keywords : Weighted principal eigenvalue problems, second order elliptic operators, singular potentials