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Groundstates and radial solutions to nonlinear Schrödinger–Poisson–Slater equations at the critical frequency

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Mercuri, Carlo [Swansea University] Moroz, Vitaly [Swansea University] Van Schaftingen, Jean [UCL]

We study the nonlocal Schr"odinger-Poisson-Slater type equation −Δu+(Iα∗|u|p)|u|p−2u=|u|q−2uin ℝN, where N∈ℕ, p>1, q>1 and Iα is the Riesz potential of order α∈(0,N). We introduce and study the Coulomb-Sobolev function space which is natural for the energy functional of the problem and we establish a family of associated optimal interpolation inequalities. We prove existence of optimizers for the inequalities, which implies the existence of solutions to the equation for a certain range of the parameters. We also study regularity and some qualitative properties of solutions. Finally, we derive radial Strauss type estimates and use them to prove the existence of radial solutions to the equation in a range of parameters which is in general wider than the range of existence parameters obtained via interpolation inequalities.

Document type Article de périodique (Journal article)
Access type Accès mixte
Publication date 2016
Language Anglais
Journal information "Calculus of Variations and Partial Differential Equations" - Vol. 55, no.6, p. 58 (2016)
Peer reviewed yes
Publisher Springer (Heidelberg)
issn 0944-2669
e-issn 1432-0835
Publication status Publié
Affiliation UCL - SST/IRMP - Institut de recherche en mathématique et physique
Keywords 1127 ; 1122 ; 1120
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Bibliographic reference Mercuri, Carlo ; Moroz, Vitaly ; Van Schaftingen, Jean. Groundstates and radial solutions to nonlinear Schrödinger–Poisson–Slater equations at the critical frequency. In: Calculus of Variations and Partial Differential Equations, Vol. 55, no.6, p. 58 (2016)
Permanent URL http://hdl.handle.net/2078.1/179187