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Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains. (English) Zbl 1340.35330
Summary: Nonlinear Schrödinger equations (NLS)\(_{a}\) with strongly singular potential \(a| x| ^{-2}\) on a bounded domain \(\Omega \) are considered. If \(\Omega =\mathbb {R}^{N}\) and \(a>-(N-2)^{2}/4\), then the global existence of weak solutions is confirmed by applying the energy methods established by N. Okazawa et al. [Evol. Equ. Control Theory 1, No. 2, 337–354 (2012; Zbl 1283.35128)]. Here \(a=-(N-2)^{2}/4\) is excluded because \(D(P_{a(N)}^{1/2})\) is not equal to \(H^{1}(\mathbb R^{N})\), where \(P_{a(N)}:=-\Delta -(N-2)^{2}/(4| x| ^{2})\) is nonnegative and selfadjoint in \(L^{2}(\mathbb R^{N})\). On the other hand, if \(\Omega \) is a smooth and bounded domain with \(0\in \Omega \), the Hardy-Poincaré inequality is proved in [J. L. Vazquez and E. Zuazua, J. Funct. Anal. 173, No. 1, 103–153 (2000; Zbl 0953.35053)]. Hence we can see that \(H_{0}^{1}(\Omega)\subset D(P_{a(N)}^{1/2}) \subset H^{s}(\Omega)\) (\(s<1\)). Therefore we can construct global weak solutions to (NLS)\(_{a}\) on \(\Omega \) by the energy methods.

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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