Classification of nonnegative solutions to static Schrödinger-Hartree and Schrödinger-Maxwell equations with combined nonlinearities

Abstract

In this paper, we are concerned with static Schrödinger-Hartree and Schrödinger-Maxwell equations with combined nonlinearities. We derive the explicit forms for positive solution $u$ in the critical case and non-existence of nontrivial nonnegative solutions in the subcritical cases (see Theorem \ref{Thm0} and \ref{Thm1}). The arguments used in our proof is a variant (for nonlocal nonlinearity) of the direct moving spheres method for fractional Laplacians in \cite{CLZ}. The main ingredients are the variants (for nonlocal nonlinearity) of the maximum principles, i.e., \emph{Narrow region principle} (Theorem \ref{Thm2} and \ref{Thm3}).

References in corpus (1)

• Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case

Cited by in corpus (5)

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• Sharp reversed Hardy-Littlewood-Sobolev inequality with extended kernel
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• Direct methods for pseudo-relativistic Schrödinger operators