Classification of minimal mass blow-up solutions for an $${L^{2}}$$ L 2 critical inhomogeneous NLS

Abstract

Abstract. We establish the classification of minimal mass blow-up solutions of the L 2 critical inhomogeneous nonlinear Schrödinger equation

Scattering for the radial 3D cubic focusing inhomogeneous nonlinear Schrödinger equation

“…where Q is the ground state of the equation −Q + ∆Q + |x| −b |Q| 4−2b N Q = 0 and 2 * = 4−2b N −2 , if N ≥ 3 or 2 * = ∞, if N = 1, 2. Also, Combet and Genoud [3] established the classification of minimal mass blow-up solutions of (1.4) with L 2 critical nonlinearity, that is, α = 4−2b N . Recently, the second author in [15], using the contraction mapping principle based on the Strichartz estimates, proved that the IVP (1.4) is locally well-posed in H 1 (R N ), for 0 < α < 2 * .…”

Small data scattering of the inhomogeneous cubic–quintic NLS in 2 dimensions

“…We say that mass and energy of solution u are conserved if they are constant w. r. t. time. The inhomogeneous NLS of single nonlinearity with coefficient behaving like |x| b (b ∈ R) have been extensively studied by the authors of [6,5,8,14,16,17,18,19,20,21,24,29,30,35]. In particular, the well-posedness for the coefficient with b > 0 has been considered under radial symmetry ( [6,5,35]).…”

Focusing nlkg equation with singular potential

“…The presence of the potential |x| −a suggests that the critical value p 0 = 1+4/(d−2) has to be shifted. It is reasonable to concentrate on the case p < p 0 (a) = 1 + (4 − 2a)/(d − 2) as in [1]. This equation has many application in nonlinear optics and in that paper a dichotomy result is established.…”

“…Here we deal with a singular potential, so that our equation is the relativistic version of the Schrödinger equation considered in [1]:…”