Existence of solutions for Schrödinger–Kirchhoff systems involving the fractional $p$-Laplacian in $\mathbb R^N$
Volume 126 / 2021
Abstract
The aim of this paper is to study the existence of weak solutions for Schrödinger–Kirchhoff systems involving the fractional $p$-Laplacian \[ \begin {cases} M([(u,v)]_{K,s,p}^p)\mathcal L_p^{s}u(x)+ V(x)|u|^{p-2}u=\lambda H_u(x, u, v)+g_1(x),\\ M([(u,v)]_{K,s,p}^p)\mathcal L_p^{s}v(x)+ V(x)|v|^{p-2}v=\lambda H_v(x, u, v)+g_2(x), \end {cases} \] in $\mathbb R^N$, where $\lambda $ is a real positive parameter, $M:[0, \infty )\to (0, \infty )$ and $V:\mathbb R^{N}\to (0,\infty )$ are continuous functions, $g_1$ and $g_2$ are perturbation terms, $\mathcal L_p^{s}$ is a nonlocal fractional operator with singular kernel $K:\mathbb R^N\setminus \{0\}\to \mathbb R^{+}$, $0 \lt s \lt 1 \lt p \lt N/s,$ and $H\in C^1(\mathbb R^{N}\times \mathbb R^2, \mathbb R)$ satisfies a non-Ambrosetti–Rabinowitz condition: there exist $\mu \in (\theta p,p_s^{*})$ and $r\ge 0$ such that \[ H_z(x,z)z-\mu H(x, z)\ge -\rho |z|^{p}-\phi (x)\quad \ \text {for all } x\in \mathbb R^N\text { and } z\in \mathbb R^2\ \text {with}\ |z|\ge r, \] where $z=(u,v)$, $H_z(x,z)=(H_u(x,u,v), H_v(x,u,v)),$ $|z|=\sqrt {u^2+v^2}$, $\theta \in [1, p_s^{*}/p)$, $\rho \ge 0$ and $0\le \phi \in L^{1}(\mathbb R^N).$ By using the Mountain Pass Theorem and Ekeland’s variational principle, we obtain the existence of solutions to the above system. Furthermore, we also investigate the existence of solutions for a system of equations with the critical exponent and the Hardy potential. Finally, we study the case that $V$ can vanish on a set of measure zero in $\mathbb R^N$ and $H$ satisfies the Ambrosetti–Rabinowitz condition.
