Pohožaev method and nontrivial ground state solutions for a class of quasilinear Schrödinger system

We study the following quasilinear Schrödinger system in the entire space \( {\mathbb {R}}^{N} \) ${\mathbb{R}}^N$ : \(\begin{aligned} -\Delta u_{j}+u_{j}+\frac{\tau }{2}\Delta (u_{j}^{2})u_{j}=\sum _{k=1}^{M} a_{j k}|u_{k}|^{p+1}|u_{j}|^{p-1} u_{j}, \quad j=1, \ldots , M,\qquad \end{aligned}\) $$\begin{array}{c}\hfill -\mathrm{\Delta }{u}_j+u_j+\frac{\tau }{2}\mathrm{\Delta }(u_j^2)u_j=\sum _{k=1}^M{a}_{\mathit{jk}}|u_k{|}^{p+1}{|u_j|}^{p-1}{u}_j,j=1,\dots ,M,\end{array}$$ where \( N\ge 3,\,0<p<\frac{2}{N-2} \) $N\ge 3,0<p<\frac{2}{N-2}$ , \( \tau >0 \) $\tau >0$ is a parameter and \(a_{j k}=a_{k j}\) $a_{\mathit{jk}}=a_{\mathit{kj}}$ are positive real numbers. Using the Pohožaev manifold approach and Moser iteration technique, we demonstrate the existence of the nontrivial ground state solutions.