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.

Cingolani, S., Gallo, M., Tanaka, K., Symmetric ground states for doubly nonlocal equations with mass constraint, <<SYMMETRY>>, 2021; 13 (7): 1-17. [doi:10.3390/sym13071199] [https://hdl.handle.net/10807/229087]

Symmetric ground states for doubly nonlocal equations with mass constraint

Gallo, Marco;
2021

Abstract

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.
2021
Inglese
Cingolani, S., Gallo, M., Tanaka, K., Symmetric ground states for doubly nonlocal equations with mass constraint, <<SYMMETRY>>, 2021; 13 (7): 1-17. [doi:10.3390/sym13071199] [https://hdl.handle.net/10807/229087]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/229087
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