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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.

We consider the following nonlinear Schrödinger equation $$\begin{aligned} -\Delta u+ V(x)u= a(x)\left| u\right| ^{q-1}u+ f(x), \quad x\in \mathbb {R}^{N}, \end{aligned}$$ -Δu+V(x)u=a(x)uq-1u+f(x),x∈RN, where Vis a non-symmetric bounded potential, ais an indefinite weight, $$0