Nonlinear Robin problems with general potential and crossing reaction

Abstract

We consider a nonlinear Robin problem driven by the $p$-Laplacian plus and indefinite potential. The reaction is parametric and exhibits an asymmetric behavior as $x\to \pm \infty$. Using variational tools together with truncation, perturbation and comparison techniques and critical groups, we show that for all small values of the parameter $\lambda >0$, the problem admits at least four nontrivial smooth solutions, three with fixed sign and the fourth nodal. By introducing a local symmetry condition on the reaction, we show that we can have a whole sequence of distinct nodal solutions converging to zero in $C^1(\overline{\Omega })$.