A degree theoretic approach for multiple solutions of constant sign for nonlinear elliptic equations (Q2476189)

The authors consider the problem \[ -\text{div}(| \nabla u| ^{p-2}\nabla u) = f(x,u) \text{ in } \Omega,\qquad u=0 \text{ on } \partial\Omega, \] where \(\Omega \subset \mathbb{R}^N\) is a bounded domain and \(f\) is a Carathéodory function. By using degree theoretic arguments based on the degree map for operators of type \((S)_+\) they prove some results concerning the existence of multiple solutions of constant sign. The hypotheses on \(f\) consider both: the case of nonresonance below the first eigenvalue, and nonresonance from above of the first eigenvalue.