Energy methods for Hartree type equations            with inverse-square potentials

Toshiyuki Suzuki

doi:10.3934/eect.2013.2.531

Article Contents

2013, Volume 2, Issue 3: 531-542. Doi: 10.3934/eect.2013.2.531

This issue Previous Article On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions Next Article On the structural properties of an efficient feedback law

Energy methods for Hartree type equations with inverse-square potentials

Toshiyuki Suzuki^1,

1.
Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601

Received: June 30, 2012

Revised: January 31, 2013

Published: August 2013

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Abstract

Nonlinear Schrödinger equations with nonlocal nonlinearities described by integral operators are considered. This generalizes usual Hartree type equations (HE)$_{0}$. We construct weak solutions to (HE)$_{a}$, $a\neq 0$, even if the kernel is of non-convolution type. The advantage of our methods is the applicability to the problem with strongly singular potential $a|x|^{-2}$ as a term in the linear part and with critical nonlinearity.

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Mathematics Subject Classification: Primary: 35Q55, 35Q40; Secondary: 81Q15.

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