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Small perturbations of critical nonlocal equations with variable exponents

Tao Lulu, He Rui, Liang Sihua

Demonstratio Mathematica

2023

Vol. 56, nr 1

art. no. 20230266

In this article, we are concerned with the following critical nonlocal equation with variable exponents: [wzór], where Ω⊂RN is a bounded domain with Lipschitz boundary, N ≥ 2 , p ∈ C (Ω×Ω) is symmetric, f : C(Ω×R)→R is a continuous function, and λ is a real positive parameter. We also assume [wzór] is the critical Sobolev exponent for variable exponents. We prove the existence of non-trivial solutions in the case of low perturbations (λ 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 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 and subcritical case.