Divergent solutions to the L 2-supercritical NLS equations

Abstract

We investigate the nonlinear Schrodinger equation iut+Δu+|u|p−1u = 0with 1+ 4/N < p < 1+ 4/N−2 (when N = 1, 2, 1 + 4/N < p < ∞) in energy space H1 and study the divergent property of infinite-variance and nonradial solutions. If \(M{\left( u \right)^{\frac{{1 - {s_C}}}{{{s_C}}}}}E\left( u \right) \prec M{\left( Q \right)^{\frac{{1 - {s_C}}}{{{s_C}}}}}E\left( Q \right)\) and \(\left\| {{u_0}} \right\|_2^{\frac{{1 - {s_c}}}{{{s_c}}}}\left\| {\nabla {u_0}} \right\|_2^{\frac{{1 - {s_c}}}{{{s_c}}}}{\left\| {\nabla Q} \right\|_2}\), then either u(t) blows up in finite forward time or u(t) exists globally for positive time and there exists a time sequence tn → +∞ such that \({\left\| {\nabla u\left( {{t_n}} \right)} \right\|_2} \to + \infty \). Here Q is the ground state solution of −(1−sc)Q+ΔQ+Qp−1Q = 0. A similar result holds for negative time. This extend the result of the 3D cubic Schrodinger equation obtained by Holmer to the general mass-supercritical and energy-subcritical case.