Dirichlet problems with unbalanced growth and double resonance

Abstract

We consider a nonlinear, nonhomogeneous Dirichlet problem driven by a nonautonomous $(p,2)$-Laplacian with unbalanced growth. The reaction is resonant both at $\pm \infty$ and at zero (double resonance). Using variational tools together with truncation and comparison techniques and critical groups, we show that the problem has at least three nontrivial bounded solutions, all with sign information (positive, negative and nodal), which are ordered.