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)su=λf(x,u)+∣u∣q(x)−2uinΩ,u=0inRN\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⊂RN\Omega \subset {{\mathbb{R}}}^{N} is a bounded domain with Lipschitz boundary, N≥2N\ge 2, p∈C(Ω×Ω)p\in C(\Omega \times \Omega ) is symmetric, f:C(Ω×R)→Rf: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∈RN:q(x)=ps∗(x)}≠∅\left\{x\in {{\mathbb{R}}}^{N}:q\left(x)={p}_{s}^{\ast }\left(x)\right\}\ne \varnothing , and ps∗(x)=Np˜(x)⁄(N−sp˜(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 ff 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=1s=1 and subcritical case.

Keywords

• fractional p-laplace operator
• p(.)-laplacian
• critical nonlinearity
• moser iteration
• concentration-compactness principles
• variational methods
• 35b33
• 35d30
• 35j20
• 46e35
• 49j35