## Existence of Multiple Non-Radial Positive Solutions of a Hénon Type Elliptic System

Jia Xiaoyao,, Lou Zhenluo,

 基金资助: 国家自然科学基金.  11571339国家自然科学基金.  11871195国家自然科学基金.  11301153河南科技大学博士启动基金.  13480051河南省高等学校重点科研项目.  20B110004

 Fund supported: NSFC.  11571339NSFC.  11871195NSFC.  11301153the Doctoral Researcher Fundation of Henan University of Science and Technology.  13480051the Key Scientific Research Projects of Higher Education Institutions in Henan Province.  20B110004

Abstract

In this paper, we study the following elliptic systemwhere $\Omega\subset\mathbb{R}^N$ is an annulus $N\geq 4$, $\mu_1, \mu_2>0$, $p, q>1$ and $p+q<\frac{2N-2}{N-3}$. By variational method and rescaling method, we prove that the system has many non-radial solutions.

Keywords： Elliptic system ; Variational method ; Non-radial solutions

Jia Xiaoyao, Lou Zhenluo. Existence of Multiple Non-Radial Positive Solutions of a Hénon Type Elliptic System. Acta Mathematica Scientia[J], 2021, 41(3): 723-728 doi:

## 1 引言

$$$\left\{\begin{array}{ll} { } -\Delta u+\mu_1 u = \frac{p}{p+q}|x|^\alpha u^{p-1}v^q, \; \; &x\in \Omega, \\ { } -\Delta v+\mu_2 v = \frac{q}{p+q}|x|^\alpha u^pv^{q-1}, \; \; &x\in \Omega, \\ u, v>0, \; \; x\in\; \Omega, \; \; u = v = 0, \; \; &x\in\partial\Omega, \end{array} \right.$$$

$$$-\Delta u = |x|^\alpha u^p, \; \; u>0, \; x\in\; \Omega, \; \; u = 0, \; x\in\; \partial \Omega,$$$

## 2 准备知识和相关引理

$\begin{eqnarray} \int_\Omega |x|^\alpha u^pv^q{\rm d}x& = &|\partial \Omega| \int_{r}^{r+d}\rho^\alpha \rho^{N-1}u^pv^q{\rm d}\rho{}\\ &\leq &C(\Omega, d)r^{N+\alpha-1}\int_{r}^{r+d}u^pv^q{\rm d}\rho{}\\ &\leq &C(\Omega, d) r^{N+\alpha-1} \bigg(\int_{r}^{r+d}u^{p+q}{\rm d}\rho\bigg)^{\frac{p}{p+q}} \bigg(\int_{r}^{r+d}v^{p+q}{\rm d}\rho\bigg)^{\frac{q}{p+q}}. \end{eqnarray}$

$\begin{eqnarray} \|(u, v)\|^{p+q} &\geq& C(\Omega, d)r^{\frac{p+q}{2}(N-1)}(a^{\frac{2}{p+q}}+b^{\frac{2}{p+q}})^{\frac{p+q}{2}}{}\\ &\geq &C(\Omega, d, p, q)r^{\frac{p+q}{2}(N-1)}(a+b). \end{eqnarray}$

$$$a^{\frac{p}{p+q}}b^{\frac{q}{p+q}}\leq \frac{p}{p+q}a+\frac{q}{p+q}b \leq \max\{\frac{p}{p+q}, \frac{q}{p+q}\} (a+b).$$$

$$$\lambda_k\leq C r^{(\frac{p+q}{2}-1)(N-2)-\alpha},$$$

选取一个非负函数$\omega(x)\in C^\infty_0(B_{\frac{d}{10}})\setminus \{0\}$, 此处$B_{\frac{d}{10}}\subset {{\Bbb R}} ^2$.

$y_1, y_2$如前面所述.事实上如果令$B(\frac{1}{\sqrt{2}}(r+\frac{d}{2}), \frac{1}{\sqrt{2}}(r+\frac{d}{2}))\in{{\Bbb R}} ^2$, 则$B$与原点的距离为$|B| = r+\frac{d}{2}$, 因此可知$(u_2(x), v_2(x))\in{\cal H}$. 综上可以得到

利用引理2.1, 可以证明$\lambda_k$是可达到的, 然后利用伸缩技巧和临界点理论得到存在$(U_k, V_k)\in\Lambda_k$为椭圆系统(1.1)的非平凡解, $k = 2, \cdots, [\frac{N}{2}]$. 利用引理3.1和3.5可知$(U_k, V_k)$是非径向对称的, 注意到引理2.2可知系统(1.1)至少存在$[\frac{N}{2}]-1$个旋转等价的非径向对称解. 综上, 完成证明.

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