Symmetric ground states for doubly nonlocal equations with mass constraint

We prove the existence of a spherically symmetric solution for a Schr\"odinger equation with a nonlocal nonlinearity of Choquard type, i.e. $$ (- \Delta)^s u + \mu u = (I_\alpha*F(u))f(u) \quad \hbox{in $\mathbb{R}^N$}, $$ where $N\geq 2$, $s \in (0,1)$, $\alpha\in (0,N)$, $I_\alpha(x)=\frac{A_{N,\alpha}}{|x|^{N-\alpha}}$ is the Riesz potential, $\mu>0$ is part of the unknowns, and $F\in C^1(\mathbb{R},\mathbb{R})$, $F' = f$ is assumed to be subcritical and to satisfy almost optimal assumptions. The mass of of the solution, described by its norm in the $L^2$-space, is prescribed in advance by $\int_{\mathbb{R}^N} u^2 \, dx = c$ for some $c>0$. The approach to this constrained problem relies on a Lagrange formulation and new deformation arguments. In addition, we prove that the obtained solution is also a ground state, which means that it realizes minimal energy among all the possible solutions to the problem.