Groundstates of the Choquard equations with a sign-changing self-interaction potential

Groundstates of the Choquard equations with a sign-changing self-interaction potential We consider a nonlinear Choquard equation $$\begin{aligned} -\Delta u+u= (V * |u|^p )|u|^{p-2}u \qquad \text {in }\mathbb {R}^N, \end{aligned}$$ - Δ u + u = ( V ∗ | u | p ) | u | p - 2 u in R N , when the self-interaction potential V is unbounded from below. Under some assumptions on $$V$$ V and on $$p$$ p , covering $$p =2$$ p = 2 and $$V$$ V being the one- or two-dimensional Newton kernel, we prove the existence of a nontrivial groundstate solution $$u\in H^1 (\mathbb {R}^N){\setminus }\{0\}$$ u ∈ H 1 ( R N ) \ { 0 } by solving a relaxed problem by a constrained minimization and then proving the convergence of the relaxed solutions to a groundstate of the original equation. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Zeitschrift für angewandte Mathematik und Physik Springer Journals

Groundstates of the Choquard equations with a sign-changing self-interaction potential

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Engineering; Theoretical and Applied Mechanics; Mathematical Methods in Physics
ISSN
0044-2275
eISSN
1420-9039
D.O.I.
10.1007/s00033-018-0975-0
Publisher site
See Article on Publisher Site

Abstract

We consider a nonlinear Choquard equation $$\begin{aligned} -\Delta u+u= (V * |u|^p )|u|^{p-2}u \qquad \text {in }\mathbb {R}^N, \end{aligned}$$ - Δ u + u = ( V ∗ | u | p ) | u | p - 2 u in R N , when the self-interaction potential V is unbounded from below. Under some assumptions on $$V$$ V and on $$p$$ p , covering $$p =2$$ p = 2 and $$V$$ V being the one- or two-dimensional Newton kernel, we prove the existence of a nontrivial groundstate solution $$u\in H^1 (\mathbb {R}^N){\setminus }\{0\}$$ u ∈ H 1 ( R N ) \ { 0 } by solving a relaxed problem by a constrained minimization and then proving the convergence of the relaxed solutions to a groundstate of the original equation.

Journal

Zeitschrift für angewandte Mathematik und PhysikSpringer Journals

Published: May 31, 2018

References

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