Combined effects for fractional Schrödinger–Kirchhoff systems with critical nonlinearities

¹ Tianjin Key Lab for Advanced Signal Processing and College of Science, Civil Aviation University of China, Tianjin, 300300, P.R. China
² Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
³ Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
⁴ Department of Mathematics, Heilongjiang Institute of Technology, Harbin, 150050, P.R. China

^* Corresponding author: zhangbinlin2012@163.com

Received: 22 October 2016
Accepted: 24 April 2017

Abstract

In this paper, we investigate the existence of solutions for critical Schrödinger–Kirchhoff type systems driven by nonlocal integro–differential operators. As a particular case, we consider the following system:

$\begin{array}{ccc}& \hfill \left\{\begin{array}{ll}M(\left[\right(u,v){]}_{s,p}^p+\Vert (u,v\left){\Vert }_{p,V}^p\right)({(-\Delta )}_p^{s}u+V(x\left)\right|u{|}^{p-2}u)=\lambda H_u(x,u,v)+\genfrac{}{}{0.1ex}{}{\alpha }{p_s^{*}}\left|v{|}^{\beta }\right|u{|}^{\alpha -2}u& \text{in}{ℝ}^N\\ M\left(\right[(u,v){]}_{s,p}^p+\Vert (u,v){\Vert }_{p,V}^p)({(-\Delta )}_p^{s}v+V(x\left)\right|u{|}^{p-2}u)=\lambda H_v(x,u,v)+\genfrac{}{}{0.1ex}{}{\beta }{p_s^{*}}\left|u{|}^{\alpha }\right|v{|}^{\beta -2}v& \text{in}{ℝ}^N,\end{array}\right\}& \hfill \end{array}$

where (–Δ)^s_p is the fractional p–Laplace operator with 0 < s < 1 < p < N/s, α, β > 1 with α + β =p^*_s, M : ℝ⁺₀ → ℝ⁺₀ is a continuous function, V : ℝ^N → ℝ⁺ is a continuous function, λ > 0 is a real parameter. By applying the mountain pass theorem and Ekeland’s variational principle, we obtain the existence and asymptotic behaviour of solutions for the above systems under some suitable assumptions. A distinguished feature of this paper is that the above systems are degenerate, that is, the Kirchhoff function could vanish at zero. To the best of our knowledge, this is the first time to exploit the existence of solutions for fractional Schrödinger–Kirchhoff systems involving critical nonlinearities in ℝ^N.