Nonlinear ellipticity on unbounded domains (Q1945736)

Let \(\Omega\subset{\mathbb R}^N\) be an unbounded open connected set. This paper is concerned with the qualitative analysis of weak solutions of the elliptic problem \[ Lu=\lambda_1\rho u-\alpha\rho u^-+\rho g(x,u)+h\qquad\text{in}\;\Omega, \] where \(Lu=-\Delta u+q(x)u\). The main result of this paper establishes that, under some natural assumptions, this problem has a weak solution \(u\in H^1_{p,q,p}(\Omega,\Gamma)\). Examples include the Schrödinger operator, the Hermite operator, and the Laguerre operator.