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## Annales Polonici Mathematici

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## Existence of solutions for Schrödinger–Kirchhoff systems involving the fractional $p$-Laplacian in $\mathbb R^N$

### Tom 126 / 2021

Annales Polonici Mathematici 126 (2021), 129-163 MSC: Primary 35D30, 35A15, 35R11, 47G20. DOI: 10.4064/ap200918-13-4 Opublikowany online: 4 June 2021

#### Streszczenie

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.

#### Autorzy

• Wei ChenChongqing University of Posts
and Telecommunications
School of Science
Chongqing, China
e-mail
• Nguyen Van ThinDepartment of Mathematics
Thai Nguyen University of Education
Thai Nguyen city, Thai Nguyen, Viet Nam
and
Thang Long Institute of Mathematics
and Applied Sciences
Thang Long University
Nghiem Xuan Yem, Hoang Mai, Hanoi, Viet Nam
e-mail
e-mail

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