\(p\)-Laplacian problems where the nonlinearity crosses an eigenvalue (Q2569359)

The authors study the following quasilinear elliptic boundary value problem \[ \begin{cases} -\text{div}(| \nabla u| ^{p-2}\nabla u) =f(x,u) &\text{in } \Omega \\ u=0&\text{on }\partial \Omega\end{cases} \tag{P} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^{n},\) \(n\geq 1\) and \(f\) is a Carathéodory function on \(\Omega \times \mathbb{R}\). Using linking arguments and a cohomological index theory they obtain non trivial solutions of (P) when \(f(x,u)\) crosses an eigenvalue. They give also a multiplicity result when \(f(x,u)\) is odd in~ \(u\).