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Groundstates and radial solutions to nonlinear Schrödinger-Poisson-Slater equations at the critical frequency. (English) Zbl 1406.35365

Summary: We study the nonlocal Schrödinger-Poisson-Slater type equation \[ - \Delta u + (I_\alpha *| u| ^p)| u| ^{p - 2} u= | u| ^{q-2}u\quad \text{in}\;\mathbb {R}^N, \] where \(N\in \mathbb {N}\), \(p>1\), \(q>1\) and \(I_\alpha\) is the Riesz potential of order \(\alpha \in (0,N).\) 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.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J47 Second-order elliptic systems
35J50 Variational methods for elliptic systems
31B35 Connections of harmonic functions with differential equations in higher dimensions
35B65 Smoothness and regularity of solutions to PDEs
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