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.
Citation
Jianqing Chen. Xiuli Tang. "Least energy solutions to an elliptic system with couplings on Kirchhoff term and nonlinear part." Adv. Differential Equations 26 (5/6) 201 - 228, May/June 2021. https://doi.org/10.57262/ade026-0506-201
Information