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Energy methods for Hartree type equations with inverse-square potentials. (English) Zbl 1282.35358
Summary: Nonlinear Schrödinger equations with nonlocal nonlinearities described by integral operators are considered. This generalizes usual Hartree type equations \((\mathbf{HE})_{0}\). We construct weak solutions to \((\mathbf{HE})_{a}\), \(a \neq 0\), 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.

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q40 PDEs in connection with quantum mechanics
81Q15 Perturbation theories for operators and differential equations in quantum theory
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