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More self-similar solutions of the nonlinear Schrödinger equation. (English) Zbl 0990.35121
Summary: We prove the existence of self-similar solutions of the Schrödinger equation in \( {\mathbb R}^N \) with power nonlinearity \( \gamma|u|^\alpha u \). Existence is obtained for a range of \( \alpha \) given by \(a< \alpha< 4/(n-4)\), \(n\geq 3\) where \((n-2)\alpha^2+ (n-4)\alpha-4=0\), which differs from, but overlaps with, the range of powers considered in [Math. Z. 228, 83-120 (1998; Zbl 0916.35109)]. We also obtain regularity results for the self-similar solutions constructed in [loc. cit].

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35D05 Existence of generalized solutions of PDE (MSC2000)
35D10 Regularity of generalized solutions of PDE (MSC2000)
35B60 Continuation and prolongation of solutions to PDEs
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