Existence of solutions to a superlinear \(p\)-Laplacian equation (Q2753142)

Existence of nontrivial solutions for the Dirichlet problem \(-\Delta_p u= f(x,u)\) in \(\Omega\), \(u=0\) on \(\partial\Omega\) is studied. Here, \(-\Delta_p u\) is the \(p\)-Laplacian, \(p>1\), and \(f\) is a Caratheodory function, which has ``superlinear'' and subcritical growth for large \(|u|\). For small \(|u|\), \(f\) is assumed to behave like \(\lambda|u|^{p-2}u\), where \(\lambda\) is between the first and the second Dirichlet eigenvalue of the \(p\)-Laplacian. The corresponding variational functional is studied by means of Morse theory.