A multiplicity theorem for problems with the \(p\)-Laplacian (Q5920430)

The existence of two nontrivial smooth solutions, one of which is of constant sign is provedbased on the degree map for operators of type \((S)_{+}\) for the following nonlinear elliptic problem: \[ -\Delta_p u=f(x,u) \quad \text{in }\Omega;\quad u=0 \quad \text{on } \partial\Omega, \] where \(\Omega\subset\mathbb{R}^n\) is bounded domain with a \(C^2\) boundary and \(1<p<+\infty.\) Here \(- \Delta_p\) denotes the negative \(p\)-Laplacian operator: \(-\Delta_p:W_0^{1,p}(\Omega)\rightarrow W^{-1,p'}(\Omega) (\frac{{1}}{p}+\frac{{1}}{p'}=1)\) which is defined by \[ \langle-\Delta_p u,\vartheta\rangle=\int_{\Omega}\| \nabla u \| ^{p-2}(\nabla u, \nabla\vartheta)_{\mathbb{R}^N} \,dx \] for all \(u,\vartheta\in W_0^{1,p}(\Omega)\).