Normalized Solutions to N-Laplacian System with Exponential Critical Growth in \(\mathbb{R}^N\)
Ruijin Zhao *
School of Mathematics and Statistics, Southwest University, Chongqing, 400715, P.R. China.
*Author to whom correspondence should be addressed.
Abstract
In this paper, we study the normalized solutions of N-Laplacian systems with exponential critical growth as follows \begin{equation}
\left\{\begin{array}{l}
-\Delta_N u=\lambda|u|^{N-2} u+G_u(u, v) \quad \text { in } \mathbb{R}^N, \\
-\Delta_N v=\mu|v|^{N-2} v+G_v(u, v) \quad \text { in } \mathbb{R}^N, \\
\int_{\mathbb{R}^N}|u|^N \mathrm{~d} x=a^N, \quad \int_{\mathbb{R}^N}|v|^N \mathrm{~d} x=b^N,
\end{array}\right.
\end{equation} where N \(\ge\) 2, a,b > 0, \(\lambda\), \(\mu\) \(\epsilon\) \(\mathbb{R}\) and the nonlinear terms Gu , Gv , the partial derivatives of the function G, have exponential critical growth in \(\mathbb{R}^N\). Using Schwarz symmetrization and Trudinger-Moser inequality, we establish the existence of normalized solutions for the above system.
Keywords: N-Laplacian system, exponential critical growth, the Trudinger-Moser inequality, normalized solutions