• 论文 •

### EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A COUPLED SYSTEM OF KIRCHHOFF TYPE EQUATIONS

Yaghoub JALILIAN

1. Department of Mathematics, Razi University, Kermanshah, Iran
• 收稿日期:2019-06-11 修回日期:2019-10-18 出版日期:2020-12-25 发布日期:2020-12-30
• 作者简介:Yaghoub JALILIAN,E-mail:y.jalilian@razi.ac.ir

### EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A COUPLED SYSTEM OF KIRCHHOFF TYPE EQUATIONS

Yaghoub JALILIAN

1. Department of Mathematics, Razi University, Kermanshah, Iran
• Received:2019-06-11 Revised:2019-10-18 Online:2020-12-25 Published:2020-12-30

Abstract: In this paper, we study the coupled system of Kirchhoff type equations \begin{equation*} \left\{ \begin{array}{ll} -\bigg(a+b\int_{\mathbb{R}^3}{|\nabla u|^{2}{\rm d}x}\bigg)\Delta u+ u = \frac{2\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta}, & x\in \mathbb{R}^3, \\[3mm] -\bigg(a+b\int_{\mathbb{R}^3}{|\nabla v|^{2}{\rm d}x}\bigg)\Delta v+ v = \frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v, & x\in \mathbb{R}^3, \\[2mm] u,v\in H^{1}(\mathbb{R}^3), \end{array} \right. \end{equation*} where $a,b > 0$, $\alpha, \beta > 1$ and $3 < \alpha+\beta < 6$. We prove the existence of a ground state solution for the above problem in which the nonlinearity is not 4-superlinear at infinity. Also, using a discreetness property of Palais-Smale sequences and the Krasnoselkii genus method, we obtain the existence of infinitely many geometrically distinct solutions in the case when $\alpha, \beta \geq 2$ and $4\leq\alpha+\beta < 6$.

• 35J50