On a superlinear elliptic \(p\)-Laplacian equation.
The authors consider the quasilinear elliptic equation \[ -\Delta_p u= f(x,u),\quad u\in W^{1,p}_0(\Omega),\tag{1} \] on a bounded domain \(\Omega\subset \mathbb{R}^N\) with smooth boundary \(\partial\Omega\); here \(\Delta_p u=\text{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplacian, \(p> 1\) and \(f:\overline\Omega\times \mathbb{R}\to \mathbb{R}\) is continuous. The goal of this paper is to find solutions of (1) inside or outside of certain order cones. In particular, they prove the existence of a sign changing solution when the nonlinearity \(f\) is superlinear and subcritical as \(|u|\to \infty\).
scientific article; zbMATH DE number 2062718
scientific article