Sign-changing solutions to an elliptic system with couplings on Kirchhoff term and nonlinear part

Abstract

We consider the following system $$ \begin{cases} \displaystyle - \Big (a_1+b_1\int_{\mathbb{R}^3} |\nabla u|^2 dx + d\int_{\mathbb{R}^3} |\nabla v|^2 dx \Big ) \Delta u + u \\ \qquad\qquad\qquad \qquad\qquad\qquad = |u|^{p-2}u +\frac{\alpha\lambda}{p}|v|^\beta |u|^{\alpha-2}u , \\[8pt] \displaystyle - \Big (a_2+b_2\int_{\mathbb{R}^3} {|\nabla v|^2} dx + d\int_{\mathbb{R}^3} |\nabla u|^2 dx \Big ) \Delta v + v \\ \qquad\qquad\qquad \qquad\qquad\qquad = |v|^{p-2}v +\frac{\beta\lambda}{p} |u|^\alpha |v|^{\beta-2}v , \end{cases} $$ where $a_1 > 0$, $a_2 > 0$; $b_1\geq 0$, $b_2\geq 0$, $d\geq 0$ and $b_1 + b_2 + d \neq 0$; $\alpha > 1$, $\beta > 1$ with $\alpha+\beta=p\in (2, 6)$ and $\lambda \geq 0$. By introducing a variant variational identity and a constraint set, we are able to prove that for any positive integer $m$, the system admits a non-radially symmetric sign-changing solution $(u,v)\in H^1(\mathbb{R}^3)\times H^1(\mathbb{R}^3)$. Moreover, $u\neq 0$, $v\neq 0$ and both $u$ and $v$ change their sign exactly $m-$times.