Least energy solutions to an elliptic system with couplings on Kirchhoff term and nonlinear part

Abstract

By establishing a new variational constraint, we prove that for $a_1 > 0$, $a_2 > 0$; $b_1\geq 0$, $b_2\geq 0$, $d\geq 0$ and $b_1 + b_2 + d \neq 0$; $\alpha > 1$, $\beta > 1$ with $ \alpha + \beta=p \in (2, 6)$; and for suitable $\lambda$, the following system $$ \left\{ \begin{aligned} - (a_1 + b_1\int_{\mathbb{R}^3} |\nabla u|^2 + d\int_{\mathbb{R}^3} |\nabla v|^2 ) \Delta u + u & = |u|^{p-2}u + \frac{\alpha\lambda}{p}|v|^\beta |u|^{\alpha-2}u , \\ -(a_2 + b_2\int_{\mathbb{R}^3} {|\nabla v|^2} + d\int_{\mathbb{R}^3} |\nabla u|^2) \Delta v + v & = |v|^{p-2}v + \frac{\beta\lambda}{p}|u|^\alpha |v|^{\beta-2}v \end{aligned} \right. $$ admits a least energy solution $(u,\ v) \in H^1(\mathbb{R}^3)\times H^1(\mathbb{R}^3)$ with $u > 0$ and $v > 0$. In the case of $a_1=a_2$ and $b_1=b_2$, several existence and nonexistence of solutions with special forms are also investigated.