Opuscula Mathematica
Opuscula Math. 46, no. 3 (2026), 405-434

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

Yuhang Tao
Jianjun Zhang
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.
Keywords: \(p\)-Laplacian system, positive normalized solution, coupled purely mass super-critical case.
Mathematics Subject Classification: 35J47, 35J62.
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About the authors ▼
Yuhang Tao
Chongqing Jiaotong University, College of Mathematics and Statistics, Xuefu, Nan'an, 400074, Chongqing, P.R. China
Jianjun Zhang (corresponding author)
Key Laboratory of Complex Systems Optimization and Intelligent Control of Chongqing Municipal Education Commission, Chongqing Jiaotong University, Chongqing, P.R. China; Chongqing Jiaotong University, College of Mathematics and Statistics, Xuefu, Nan'an, 400074, Chongqing, P.R. China
About the article ▼
Communicated by Daniele Cassani.
Received: 2026-01-27.
Revised: 2026-03-25.
Accepted: 2026-03-31.
Published online: 2026-06-15.
Cite this article as ▼
Yuhang Tao, Jianjun Zhang, Normalized ground states for a p-Laplacian system in the mass super-critical case, Opuscula Math. 46, no. 3 (2026), 405-434, https://doi.org/10.7494/OpMath.202603311.
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