Normalized solutions to the Schrödinger systems with double critical growth and weakly attractive potentials

Abstract

In this paper, we look for solutions to the following critical Schrödinger system $$\begin{cases} -\Delta u+(V_1+\lambda_1)u=|u|^{2^*-2}u+|u|^{p_1-2}u+\beta r_1|u|^{r_1-2}u|v|^{r_2}&{\rm in}\ \mathbb{R}^N,\\ -\Delta v+(V_2+\lambda_2)v=|v|^{2^*-2}v+|v|^{p_2-2}v+\beta r_2|u|^{r_1}|v|^{r_2-2}v&{\rm in}\ \mathbb{R}^N, \end{cases}$$ having prescribed mass $\int_{\mathbb{R}^N}u^2=a_1>0$ and $\int_{\mathbb{R}^N}v^2=a_2>0$, where $\lambda_1,\lambda_2\in\mathbb{R}$ will arise as Lagrange multipliers, $N\geqslant3$, $2^*=2N/(N-2)$ is the Sobolev critical exponent, $r_1,r_2>1$, $p_1,p_2,r_1+r_2\in(2+4/N,2^*)$ and $\beta>0$ is a coupling constant. Under suitable conditions on the potentials $V_1$ and $V_2$, $\beta_*>0$ exists such that the above Schrödinger system admits a positive radial normalized solution when $\beta\geqslant\beta_*$. The proof is based on comparison argument and minmax method.

Keywords

• schrödinger systems
• weakly attractive potentials
• normalized solutions
• positive solutions