Existence of solutions to a class of quasilinear Schrödinger systems involving the fractional $p$-Laplacian - Mingqi Xiang; Binlin Zhang; Zhe Wei (Electronic Journal of Qualitative Theory of Differential Equations, 2016)
The purpose of this paper is to investigate the existence of solutions to the following quasilinear Schr\"{o}dinger type system driven by the fractional $p$-Laplacian \begin{align*} (-\Delta)^{s}_pu+a(x)|u|^{p-2}u&=H_u(x,u,v)\quad \mbox{in } \mathbb{R}^N,\\ (-\Delta)^s_qv+b(x)|v|^{q-2}v&=H_v(x,u,v)\quad \mbox{in } \mathbb{R}^N, \end{align*} where $1<q\leq p$, $sp<N$, $(-\Delta )_m^s$ is the fractional $m$-Laplacian, the coefficients $a, b$ are two continuous and positive functions, and $H_u,H_v$ denote the partial derivatives of $H$ with respect to the second variable and the third variable. By using the mountain pass theorem, we obtain the existence of nontrivial and nonnegative solutions for the above system. The main feature of this paper is that the nonlinearities do not necessarily satisfy the Ambrosetti-Rabinowitz condition.