Normalized ground states for a $p$-Laplacian system in the mass super-critical case

Abstract

In this paper, we study the existence of positive normalized solutions to the following $p$-Laplacian system: $\begin{cases} -\Delta_p u+\lambda_1u^{p-1}=\mu_1u^{m_1-1}+\beta r_1u^{r_1-1}v^{r_2}&\text{in }\mathbb{R}^N,\\ -\Delta_p v+\lambda_2v^{p-1}=\mu_2v^{m_2-1}+\beta r_2u^{r_1}v^{r_2-1}&\text{in }\mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u|^p=a, \quad \int_{\mathbb{R}^N}|v|^p=b,\end{cases}$ where $1\lt p\lt N$, $\mu_1,\mu_2,\beta,a,b\gt 0$ are prescribed, $\lambda_1,\lambda_2 \in \mathbb{R}$ are known as the Lagrange multiplier, $\Delta_p u= \mathrm{div} (|\nabla u|^{p-2} \nabla u)$ denotes the $p$-Laplacian operator. We prove the existence of positive solutions for the coupled purely mass super-critical case (i.e., $\frac{p^2}{N}+p\lt m_1,m_2,r_1 + r_2\lt p^*$) by a minimization argument based on a closed ball and the Pohozaev constraint.