Normalized Ground State Solution to a Mixed Schrödinger System in a Plane

Abstract

In this paper, we establish the existence of positive ground state solutions for a class of mixed Schrödinger systems with concave-convex nonlinearities in $\mathbb{R}^2$, that is \begin{cases} -\partial_{xx}u + (-\Delta)_y^s u + \lambda_1 u = \mu_1 u^{p-1} + \beta r_1 u^{r_1-1}v^{r_2}, \\ -\partial_{xx}v + (-\Delta)_y^s v + \lambda_2 v = \mu_2 v^{q-1} + \beta r_2 u^{r_1}v^{r_2-1} \end{cases} subject to the $L^2$-norm constraints \begin{equation}
\int_{\mathbb{R}^{2}} u^{2}dxdy = a, \qquad \int_{\mathbb{R}^{2}} v^{2}dxdy = b,
\end{equation}
where $(x,y)\in\mathbb{R}^2$, $u,v\geq0$, $s\in(1/2,1)$, $\mu_1,\mu_2,\beta>0$, $r_1,r_2>1$, the prescribed masses $a,b>0$, and the parameters $\lambda_1,\lambda_2$ appear as Lagrange multipliers. Moreover, the exponents $p,q,r_1+r_2$ satisfy $2(1+3s)/(1+s) < p,q,r_1+r_2 < 2_s$, where $2_s = 2(1+s)/(1-s)$. To obtain our main existence results, we employ variational techniques such as the mountain pass theorem, the Pohozaev manifold, Steiner rearrangement, and others, consolidating the works [Jeanjean et al., Nonlinear Differ. Equ. Appl. 31 (2024)].