Infinitely Many Sign-Changing Solutions for Coupled Schrödinger System

Abstract

In this article, we study the following coupled Schr $\ddot{\mathbf{o}}$ dinger system with subcritical exponent: $$\{\begin{array}{l}-\mathrm{\Delta }u+\alpha u=\mu |u{|}^{p-1}u+\lambda |u{|}^{\frac{p-3}{2}}u|v{|}^{\frac{p+1}{2}},x\in \mathrm{\Omega },\\ -\mathrm{\Delta }v+\beta v=\nu |v{|}^{p-1}v+\lambda |u{|}^{\frac{p+1}{2}}|v{|}^{\frac{p-3}{2}}v,x\in \mathrm{\Omega },\\ u=v=0,x\in \mathrm{\partial }\mathrm{\Omega }.\end{array}$$ Here, either $\mathrm{\Omega }={\mathbb{R}}^N$ or $\mathrm{\Omega }$ is a smooth bounded domain in ${\mathbb{R}}^N$ with $N\ge 4$ ; positive constants $\alpha ,\beta ,\mu ,\nu >0$ and the coupling constant $\lambda <0$ , $1<p<2^{*}-1$ , $2^{*}=2N/N-2$ is the critical Sobolev exponent. We show that, this system has infinitely many radially symmetric sign-changing solutions and infinitely many radially symmetric semi-nodal solutions. Moreover, we obtain a least energy sign-changing solution for $\alpha >-{\lambda }_1(\mathrm{\Omega }),\beta >-{\lambda }_1(\mathrm{\Omega })$ in the following sense: one component is positive and the other one changes sign which has exactly two nodal domains.