Small perturbations of critical nonlocal equations with variable exponents

Abstract In this article, we are concerned with the following critical nonlocal equation with variable exponents: ( − Δ ) p ( x , y ) s u = λ f ( x , u ) + ∣ u ∣ q ( x ) − 2 u in Ω , u = 0 in R N \ Ω , \left\{\begin{array}{ll}{\left(-\Delta )}_{p\left(x,y)}^{s}u=\lambda f\left(x,u)+{| u| }^{q\left(x)-2}u& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=0& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}\backslash \Omega \right,\end{array}\right. where Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} is a bounded domain with Lipschitz boundary, N ≥ 2 N\ge 2 , p ∈ C ( Ω × Ω ) p\in C(\Omega \times \Omega ) is symmetric, f : C ( Ω × R ) → R f:C\left(\Omega \times {\mathbb{R}})\to {\mathbb{R}} is a continuous function, and λ \lambda is a real positive parameter. We also assume that { x ∈ R N : q ( x ) = p s ∗ ( x ) } ≠ ∅ \left\{x\in {{\mathbb{R}}}^{N}:q\left(x)={p}_{s}^{\ast }\left(x)\right\}\ne \varnothing , and p s ∗ ( x ) = N p ˜ ( x ) ⁄ ( N − s p ˜ ( x ) ) {p}_{s}^{\ast }\left(x)=N\tilde{p}\left(x)/\left(N-s\tilde{p}\left(x)) is the critical Sobolev exponent for variable exponents. We prove the existence of non-trivial solutions in the case of low perturbations ( λ \lambda small enough) by using the mountain pass theorem, the concentration-compactness principles for fractional Sobolev spaces with variable exponents, and the Moser iteration method. The features of this article are the following: (1) the function f f does not satisfy the usual Ambrosetti-Rabinowitz condition and (2) this article contains the presence of critical terms, which can be viewed as a partial extension of the previous results concerning the the existence of solutions to this problem in the case of s = 1 s=1 and subcritical case.