Multiple positive solutions for a class of pp-superlinear elliptic problems

We study nonlinear elliptic Dirichlet problems driven by the pp-Laplacian and with a nonlinearity which consists of a pp-superlinear term perturbed by a pp-linear one. Using critical point theory, coupled with suitable truncation techniques, we show that the problem has two positive smooth solutions. When the conditions are symmetric in the two semiaxes, we have four nontrivial smooth solutions of constant sign. Finally for the semilinear problem, using Morse theory, we produce a fifth nontrivial smooth solution.