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Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. (English) Zbl 1285.35048

Summary: We consider a semilinear elliptic problem \[ -{\Delta}u + u = (I_{\alpha} \ast |u|^p)|u|^{p-2}u \quad \text{in } \mathbb R^N, \] where \(I_{\alpha}\) is a Riesz potential and \(p > 1\). This family of equations includes the Choquard or nonlinear Schrödinger-Newton equation. For an optimal range of parameters we prove the existence of a positive groundstate solution of the equation. We also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point. Finally, we derive the decay asymptotics at infinity of the groundstates.

MSC:

35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
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