Multiplicity results for nonlocal critical elliptic problems

Abstract

We prove new multiplicity results for some nonlocal critical growth elliptic problems in bounded domains. More specifically, we show that the problems considered here have arbitrarily many solutions for all sufficiently large values of a certain parameter $\lambda >0$. In particular, the number of solutions goes to infinity as $\lambda \to \infty$. We also give an explicit lower bound on $\lambda$ in order to have a given number of solutions. This lower bound is in terms of a sequence of eigenvalues of the associated nonlocal elliptic operator. The proofs are based on an abstract critical point theorem.