Nonlinear, nonhomogeneous Robin problems with indefinite potential and general reaction

TL;DR

This paper studies nonlinear elliptic equations with indefinite potential and general reaction terms, using variational methods to find multiple solutions, including nodal solutions, under various conditions.

Contribution

It introduces new variational techniques to establish multiple solutions for complex nonlinear elliptic problems with indefinite potentials.

Findings

Produced three nontrivial solutions with sign information.

Obtained a second nodal solution in the semilinear case, totaling four solutions.

Generated an infinite sequence of nodal solutions under symmetry conditions.

Abstract

We consider a nonlinear elliptic equation driven by a nonhomogeneous differential operator plus an indefinite potential. On the reaction term we impose conditions only near zero. Using variational methods, together with truncation and perturbation techniques and critical groups, we produce three nontrivial solutions with sign information. In the semilinear case we improve this result by obtaining a second nodal solution for a total of four nontrivial solutions. Finally, under a symmetry condition on the reaction term, we generate a whole sequence of distinct nodal solutions.