Remarks on sublinear elliptic problems on unbounded cylinders (Q1922913)

The authors consider the problem \[ -\Delta u+\varrho(x)u=f(x,u)\quad \text{in}\quad\Omega,\quad u\geq 0,\;u\neq 0,\;\int_\Omega\varrho(x)u^2dx<\infty,\tag{\(*\)} \] where \(\Omega=D\times\mathbb{R}^m\), \(D\subset\mathbb{R}^\ell\) is a bounded smooth domain (\(m,\ell\geq 0\)) and \(f:\Omega\times[0,\infty)\to\mathbb{R}\) is a Carathéodory function, \(\varrho:\Omega\to\mathbb{R}\) is continuous, \(\varrho(x)\geq \varrho_0>0\), \(\varrho(x)\to\infty\) as \(|x|\to\infty\). Under growth conditions on \(f(x,s)\) and its potential \(F(x,t)=\int^t_0 f(x,s)ds\) it is shown that problem \((*)\) has a solution. The assumptions are discussed and special problems \((*)\) are presented.