Nonlinear Robin problems with locally defined reaction

Abstract

We consider a nonlinear Robin problem driven by a p− Laplacian. The reaction consistes of two terms. The first one is parametric and only locally defined, while the second one is (p − 1)- superlinear. Using cutt-off techniques together with critical point theory and critical groups, we show that for big values of the parameter λ > 0, the problem has at least three nontrivial solutions, all with sign information (positive, negative and nodal). In the semilinear case (p = 2), we produce a second nodal solution, for a total of four nontrivial solutions, all with sign information.