Normalized solution for a kind of coupled Kirchhoff systems

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Abstract

In this paper, we investigate the existence of a normalized solution for the following Kirchhoff system in the entire space $\mathbb{R}^{N}(N \geq 3)$:

(P)$\left\{\begin{array}{l} -\left(1+\int_{\mathbb{R}^{N}}|\nabla u|^{2} d x\right) \Delta u=\lambda_{1} u+\mu_{1}|u|^{p-2} u+\beta r_{1}|u|^{r_{1}-2} u|v|^{r_{2}}, \\ -\left(1+\int_{\mathbb{R}^{N}}|\nabla v|^{2} d x\right) \Delta v=\lambda_{2} v+\mu_{2}|v|^{q-2} v+\beta r_{2}|u|^{r_{1}}|v|^{r_{2}-2} v, \end{array}\right.$

under the constraints $\int_{\mathbb{R}^{N}}|u|^{2} d x=m_{1}$ and $\int_{\mathbb{R}^{N}}|v|^{2} d x=m_{2}$, where m₁, m₂ > 0 are prescribed. The parameters $\mu_{1}, \mu_{2}, \beta>0,2 \leq p, q<2+\frac{8}{N}, r_{1}, r_{2}>$, and satisfy $r_{1}+r_{2}=2^{*}=\frac{2 N}{N-2}$. The frequencies λ₁, λ₂ appear as Lagrange multipliers. With the help of the Pohožaev manifold and the minimization of the energy functional over a combination of the mass constraints and the closed balls, we obtain a positive ground state solution to (P). We mainly extend the results of Yang (Normalized ground state solutions for Kirchhoff-type systems) concerning the above problem from a single critical to a coupled critical nonlinearity.

Keywords

normalized solutions / nonlinearity / coupled Kirchhoff equation / Pohožaev manifold

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