A Critical Point Theorem for Perturbed Functionals and Low Perturbations of Differential and Nonlocal Systems

Abstract

In this paper we establish a new critical point theorem for a class of perturbed differentiable functionals without satisfying the Palais–Smale condition. We prove the existence of at least one critical point to such functionals, provided that the perturbation is sufficiently small. The main abstract result of this paper is applied both to perturbed nonhomogeneous equations in Orlicz–Sobolev spaces and to nonlocal problems in fractional Sobolev spaces.

Keywords

• critical point theorem
• perturbed functional
• orlicz–sobolev space
• fractional sobolev space
• existence of solutions
• 35j70
• 35p30
• 76h05