Sign-changing solutions for coupled Schrödinger system

Abstract

In this paper we study the following nonlinear Schrödinger system: $$ \textstyle\begin{cases} -\Delta u+\alpha u = \vert u \vert ^{p-1}u+\frac{2}{q+1} \lambda \vert u \vert ^{ \frac{p-3}{2}}u \vert v \vert ^{\frac{q+1}{2}},\quad x \in \mathbb{R}^{3}, \\ -\Delta v+\beta v = \vert v \vert ^{q-1}v+\frac{2}{p+1} \lambda \vert u \vert ^{ \frac{p+1}{2}} \vert v \vert ^{\frac{q-3}{2}}v ,\quad x \in \mathbb{R}^{3}, \\ u(x)\rightarrow 0,\qquad v(x)\rightarrow 0,\quad \text{as } \vert x \vert \rightarrow \infty , \end{cases} $$ where $3\leq p, q<5$ , α, β are positive parameters. We show that there exists $\lambda _{k}>0$ such that the equation has at least k radially symmetric sign-changing solutions and at least k seminodal solutions for each $k\in \mathbb{N}$ and $\lambda \in (0, \lambda _{k})$ . Moreover, we show the existence of a least energy radially symmetric sign-changing solution for each $\lambda \in (0, \lambda _{0})$ where $\lambda _{0}\in (0, \lambda _{1}]$ .