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

We consider a nonlinear elliptic problem driven by the pp-Laplacian, where the right hand side nonlinearity exhibits a pp-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 pp-Laplacian, due to Perera and Dancer–Perera.