Asymptotic and limiting profiles of blowup solutions of the nonlinear Schr�dinger equation with critical power

Abstract

This paper is a sequel to previous ones 38, 39, 41. We continue the study of the blowup problem for the nonlinear Schrodinger equation with critical power nonlinearity (NSC). We introduce a new idea to prove the existence of a blowup solution in H1(ℝN) without any weight condition and reduce the problem to a kind of variational problem. Our new method refines the previous results concerning the asymptotic and limiting profiles of blowup solutions: For a certain class of initial data, the blowup solution behaves like a finite superposition of zero-energy, H1-bounded, global-in-time solutions of (NSC); these singularities stay in a bounded region in ℝN, and one can see that the so-called shoulder emerges outside these singularities as suggested by some numerical computations (see, e.g., [26]). We investigate the asymptotic behavior of zero-energy, global-in-time solutions of (NSC) and find that such a solution behaves like a “multisoliton.” However, it is not an assemblage of free “particles”; the “solitons” interact with each other. © 1999 John Wiley & Sons, Inc.