Scattering for the radial focusing inhomogeneous NLS equation in higher dimensions. (English) Zbl 1437.35623
Summary: We consider the inhomogeneous nonlinear Schrödinger equation \[ i u_t +\Delta u+|x|^{-b}|u|^\alpha u = 0, \quad x\in \mathbb{R}^N,\] where \(N\geq 2\), \(\frac{4-2b}{N}<\alpha <\frac{4-2b}{N-2}\) (\(\frac{4-2b}{N}<\alpha <\infty\) if \(N=2)\) and \(0<b<\min \{N/3,1\} \). For a radial initial data \(u_0 \in H^1({\mathbb{R}}^N)\), under a certain smallness condition, we prove that the corresponding solution is global and scatters. The smallness condition is related to the ground state solution of \(-Q+\Delta Q+ |x|^{-b}|Q|^{\alpha }Q=0\) and the critical Sobolev index \(s_c=\frac{N}{2}-\frac{2-b}{\alpha } \). This is an extension of the recent work [L. G. Farah and C. M. Guzmán, J. Differ. Equations 262, No. 8, 4175–4231 (2017; Zbl 1362.35284)] by the same authors, where they consider the case \(N=3\) and \(\alpha =2\). The proof is inspired by the concentration-compactness/rigidity method developed by C. E. Kenig and F. Merle [Invent. Math. 166, No. 3, 645–675 (2006; Zbl 1115.35125)] to study the \(H^1({\mathbb{R}}^N)\) critical NLS equation and also J. Holmer and S. Roudenko [Commun. Math. Phys. 282, No. 2, 435–467 (2008; Zbl 1155.35094)] in the case of \(H^1({\mathbb{R}}^N)\) subcritical and \(L^2({\mathbb{R}}^N)\) supercritical NLS equations.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 78A60 | Lasers, masers, optical bistability, nonlinear optics |
| 78A45 | Diffraction, scattering |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
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