Existence of three nontrivial solutions for asymptotically p-linear noncoercive p-Laplacian equations

Abstract

Weconsider a nonlinear elliptic problem driven by the p-Laplacian, where the right hand side nonlinearity exhibits a p-linear behavior near infinity and the Euler functional of the problem need not be coercive and, in fact, can be indefinite. Using a combination of minimax arguments with truncation techniques and Morse theory, we show that the problem has at least three nontrivial smooth solutions, two of which have a constant sign. Our method of proof uses some results on critical groups and the spectrum of the p-Laplacian, due to Perera and Dancer–Perera.