We study the nonlocal Schrödinger–Poisson–Slater type equation -Δu+(Iα ∗ |u|p)|u|p-2u=|u|q-2u in 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.

Groundstates and radial solutions to nonlinear Schrödinger–Poisson–Slater equations at the critical frequency / Mercuri, C.; Moroz, V.; Van Schaftingen, J.. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - 55:6(2016), pp. 1-58. [10.1007/s00526-016-1079-3]

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

Mercuri C.;
2016

Abstract

We study the nonlocal Schrödinger–Poisson–Slater type equation -Δu+(Iα ∗ |u|p)|u|p-2u=|u|q-2u in 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.
2016
55
6
1
58
Groundstates and radial solutions to nonlinear Schrödinger–Poisson–Slater equations at the critical frequency / Mercuri, C.; Moroz, V.; Van Schaftingen, J.. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - 55:6(2016), pp. 1-58. [10.1007/s00526-016-1079-3]
Mercuri, C.; Moroz, V.; Van Schaftingen, J.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/1295812
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