Sign-changing solutions for coupled Schrödinger system

Abstract

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