Existence of weak solutions for degenerate semilinear elliptic equations in unbounded domains

In this study, we prove the existence of a weak solution for the degenerate semilinear elliptic Dirichlet boundary-value problem $$displaylines{ Lu-mu u g_{1} + h(u) g_{2}= fquad hbox{in }Omega,cr u = 0quad hbox{on }partialOmega }$$ in a suitable weighted Sobolev space. Here the domain $Omegasubsetmathbb{R}^{n}$, $ngeq 3$, is not necessarily bounded, and $h$ is a continuous bounded nonlinearity. The theory is also extended for $h$ continuous and unbounded.