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

We study the nonlocal Schrödinger–Poisson–Slater type equation - Δ u + ( I α ∗ | u | p ) | u | p - 2 u = | u | q - 2 u in R N , where N ∈ N , p > 1 , q > 1 and I α is the Riesz potential of order α ∈ ( 0 , N ) . We introduce and study the Coulomb–Sobolev function space which is natural for the...

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Bibliographic Details
Published in:Calculus of variations and partial differential equations 2016-12, Vol.55 (6), p.1-58, Article 146
Main Authors: Mercuri, Carlo, Moroz, Vitaly, Van Schaftingen, Jean
Format: Article
Language:English
Subjects:
Analysis
Calculus of Variations and Optimal Control
Optimization
Control
Function space
Inequalities
Interpolation
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Nonlinear equations
Parameters
Systems Theory
Theoretical
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Summary:We study the nonlocal Schrödinger–Poisson–Slater type equation - Δ u + ( I α ∗ | u | p ) | u | p - 2 u = | u | q - 2 u in R N , where N ∈ N , p > 1 , q > 1 and I α 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.
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-016-1079-3