Groundstates of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent

By Vitaly Moroz, Jean Van Schaftingen. 2014

228 citations recorded by the source corpus.

We consider nonlinear Choquard equation where N ≥ 3, V ∈ L∞(ℝN) is an external potential and Iα(x) is the Riesz potential of order α ∈ (0, N). The power in the nonlocal part of the equation is critical with respect to the Hardy–Littlewood–Sobolev inequality. As a consequence, in the associated minimization problem a loss of compactness may occur. We prove that if then the equation has a nontrivial solution. We also discuss some necessary conditions for the existence of a solution. Our considerations are based on a concentration compactness argument and a nonlocal version of Brezis–Lieb lemma.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c875584a81e6899ad076289744c914dcc0ac7a2b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Fashun Gao
2. Minbo Yang
3. Marco Squassina
4. Claudianor Oliveira Alves
5. Jianjun Zhang
6. Daniele Cassani
7. Giovany M. Figueiredo
8. Vicentiu D. Rădulescu

In Mathematical Physics

1. Fashun Gao
2. Minbo Yang
3. Claudianor Oliveira Alves
4. Marco Squassina
5. Jean Van Schaftingen
6. Xinfu Li

Citations in the corpus

• Singularly perturbed critical Choquard equations (2016, background): “The authors also considered in [25,27] the existence of ground states under the assumption of Berestycki–Lions type and studied the existence of solutions for the nonlocal equation with lower critical exponent due to the Hardy–Littlewood–Sobolev inequality.”