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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.

MSC:
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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