Blow-up solutions for the inhomogeneous Schrödinger equation with \(L^2\) supercritical nonlinearity. (English) Zbl 1306.35124
Summary: This paper studies blow-up solutions for the inhomogeneous Schrödinger equation with \(L^2\) supercritical nonlinearity. In terms of Strauss’ arguments in [W. A. Strauss, Commun. Math. Phys. 55, 149–162 (1977; Zbl 0356.35028)], we find a new compactness lemma for radial symmetric functions. Thus, we use it to derive the best constants of two generalized Gagliardo-Nirenberg type inequalities. Moreover, we obtain a more precisely sharp criteria of blow-up and global existence, and derive the weak concentration phenomenon of blow-up solutions by the variational methods.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35B44 | Blow-up in context of PDEs |
| 35A23 | Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals |
Keywords:
nonlinear inhomogeneous Schrödinger equation; blow-up solution; generalized Gagliardo-Nirenberg inequality; weak concentrationCitations:
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