Existence of normalized ground state solution to a mixed Schrödinger system in a plane

In this paper, we establish the existence of positive ground state solutions for a class of mixed Schrödinger systems with concave-convex nonlinearities in $\mathbb{R}^2$, subject to $L^2$-norm constraints; that is, $\left\{ \begin{aligned} -\partial_{xx} u + (-Δ)_y^s u + λ_1 u &= μ_1 u^{p-1} + βr_1 u^{r_1-1} v^{r_2}, && -\partial_{xx} v + (-Δ)_y^s v + λ_2 v &= μ_2 v^{q-1} + βr_2 u^{r_1} v^{r_2-1}, && \end{aligned} \right.$ subject to the $L^2$-norm constraints: $\int_{\mathbb{R}^2} u^2 \,\mathrm{d}x\mathrm{d}y = a \quad \text{and} \quad \int_{\mathbb{R}^2} v^2 \,\mathrm{d}x\mathrm{d}y = b,$ where $(x,y)\in \mathbb{R}^2$, $u, v \geq 0$, $s \in \left(1/2, 1 \right)$, $μ_1, μ_2, β> 0$, $r_1, r_2 > 1$, the prescribed masses $a, b > 0$, and the parameters $λ_1, λ_2$ appear as Lagrange multipliers. Moreover, the exponents $p, q, r_1 + r_2$ satisfy: $\frac{2(1+3s)}{1+s} < p, q, r_1 + r_2 < 2_s,$ where $2_s = \frac{2(1+s)}{1-s}$. 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 of Louis Jeanjean et al. \cite{jeanjean2024normalized}.

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