Multiple constant sign and nodal solutions for nonlinear nonhomogeneous elliptic equations depending on a parameter

Abstract

We study a parametric nonlinear Dirichlet problem driven by a nonhomogeneous differential operator and with a reaction which has z-dependent zeros of constant sign. For all big values of the parameter $$\lambda >0$$ , we prove two multiplicity theorems producing two positive solutions, two negative solutions, and two nodal solutions. In the first we do not impose any asymptotic conditions on the reaction $$f(z,\cdot )$$ at zero. In the second we do not impose any asymptotic conditions on the reaction $$f(z,\cdot )$$ at $$\pm \infty $$ . Then we produce a total of twelve nontrivial smooth solutions all with sign information. Our proofs use variational methods together with flow invariance arguments and suitable truncation techniques.