Abstract
We prove the existence of a spherically symmetric solution for a Schrödinger equation with a nonlocal nonlinearity of Choquard type. This term is assumed to be subcritical and satisfy almost optimal assumptions. The mass of of the solution, described by its norm in the Lebesgue space, is prescribed in advance. 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.
Highlights
In 1954, the nonlocal equationAcademic Editor: Alexander Shapovalov − ∆u + μu = 1 4π|x| ∗ |u|2 u in R3 (1)Received: 31 May 2021 Accepted: 25 June 2021 Published: 2 July 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.was introduced by Pekar [1] in the framework of quantum mechanics, and in 1976 it arose in the work of Choquard on the modeling of an electron trapped in its own hole, in a certain approximation to the Hartree–Fock theory of one-component plasma [2]
Was introduced by Pekar [1] in the framework of quantum mechanics, and in 1976 it arose in the work of Choquard on the modeling of an electron trapped in its own hole, in a certain approximation to the Hartree–Fock theory of one-component plasma [2]
For the general class of nonlinearities of the Berestycki–Lions type [18,54], satisfying (f1)–(f4), we introduce a Lagrangian formulation in order to obtain L2-normalized solutions of the nonlocal problem (2) in the spirit of [42], where it is applied for fractional NLS
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. The existence and qualitative properties of the solutions for more general classes of fractional NLS equations with local source were studied in [38,39,40,41]. For the general class of nonlinearities of the Berestycki–Lions type [18,54], satisfying (f1)–(f4), we introduce a Lagrangian formulation in order to obtain L2-normalized solutions of the nonlocal problem (2) in the spirit of [42], where it is applied for fractional NLS equations with a local source (see [55]). We recall that C2-solutions to (6) satisfy the Pohozaev identity (see ([53], Proposition 2) and ([49], Equation (6.1))) Inspired by this identity, we introduce the Pohozaev functional P : R+ × Hrs(RN) → R by setting.
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