Existence and multiplicity of standing wave solutions for perturbed fractional p-Laplacian systems involving critical exponents

Abstract

In this paper, we investigate the existence of standing wave solutions to the following perturbed fractional p-Laplacian systems with critical nonlinearity $ \begin{equation*} \left\{ \begin{aligned} &\varepsilon^{ps}(-\Delta)^{s}_{p}u + V(x)|u|^{p-2}u = K(x)|u|^{p^{*}_{s}-2}u + F_{u}(x, u, v), \; x\in \mathbb{R}^{N}, \\ &\varepsilon^{ps}(-\Delta)^{s}_{p}v + V(x)|v|^{p-2}v = K(x)|v|^{p^{*}_{s}-2}v + F_{v}(x, u, v), \; x\in \mathbb{R}^{N}. \end{aligned} \right. \end{equation*} $ Under some proper conditions, we obtain the existence of standing wave solutions $ (u_{\varepsilon}, v_{\varepsilon}) $ which tend to the trivial solutions as $ \varepsilon\rightarrow 0 $. Moreover, we get $ m $ pairs of solutions for the above system under some extra assumptions. Our results improve and supplement some existing relevant results.

Keywords

• perturbed fractional p-laplacian systems
• critical growth
• variational methods
• standing wave solutions