Abstract:

In this article, we study the existence of sign-changing and multiple solutions for the following mixed order nonlinear elliptic system

$\begin{equation} \left\{ \begin{aligned} {(-\Delta)}^s{u}&=|u|^{3s-1}u+\beta v^2|u|^{\alpha-1}u, && {\rm in}\ \Omega,\\ -\Delta {v}&=|v|^{2}v+\frac{2\beta}{\alpha+1} |u|^{\alpha }uv, && {\rm in}\ \Omega,\\ u=v&=0, && {\rm on}\ \partial\Omega,\nonumber \end{aligned} \right. \end{equation}$

where $\Omega\subset \mathbb{R}^n(2\leq n \leq3 )$ is a smooth bounded domain, the parameters $ \beta<0,\ \alpha>0$. $(-\Delta )^s$ is a fractional Laplacian operator, and $-\Delta $ is a Laplacian operator. If $s\in[\frac{3}{4},\ 1)$ and $\alpha\in(0,\frac{6s-n}{n-2s})$, we prove the existence of positive, negative, and sign-changing solutions for the above system. Moreover, if $\alpha=\frac{n_{1}}{m_{1}}, s=\frac{n_{2}}{m_{2}}, m_{1}, n_{1}, m_{2}, n_{2}$ are odd integers, then there exists an unbounded sequence of sign-changing solutions $\{(u_m,v_m)\}, m\in\mathbb{N}$, and $u_m, v_m$ both have at most $m+1$ sign-changing domains.

Key words: mixed order, fractional Laplacian, nonlinear elliptic system, sign-changing solution