We consider L 2 -supercritical and H 1 -subcritical focusing nonlinear Schrödinger equations. We introduce a subset P W of H 1 (R d ) for d ≥ 1, and investigate behavior of the solutions with initial data in this set. For this end, we divide P W into two disjoint components P W + and P W − . Then, it turns out that any solution starting from a datum in P W + behaves asymptotically free, and solution starting from a datum in P W − blows up or grows up, from which we find that the ground state has two unstable directions. We also investigate some properties of generic global and blowup solutions.It is worth while noting the following facts: 1. P W + and P W − are unbounded open sets in H 1 (R d ): Indeed, one can easily verify this fact by considering the scaled functions f λ (x) := λ 2. P W = P W + ∪ P W − and P W + ∩ P W − = ∅ (see Lemma 2.1) 3. P W + and P W − are invariant under the flow defined by the equation (1.1) (see Proposition 2.5 and Proposition 2.6).

We study the energy-critical focusing nonlinear Schrödinger equation with an energysubcritical perturbation. We show the existence of a ground state in the four or higher dimensions. Moreover, we give a sufficient and necessary condition for a solution to scatter, in the spirit of Kenig-Merle [16].

The study of the uniqueness and nondegeneracy of ground state solutions to semilinear elliptic equations is of great importance because of the resulting energy landscape and its implications for the various dynamics. In [2], semilinear elliptic equations with combined power-type nonlinearities involving the Sobolev critical exponent are studied. There, it is shown that if the dimension is four or higher, and the frequency is sufficiently small, then the positive radial ground state is unique and nondegenerate. In this paper, we extend these results to the case of high frequencies when the dimension is five and higher. After suitably rescaling the equation, we demonstrate that the main behavior of the solutions is given by the Sobolev critical part for which the ground states are explicit, and their degeneracy is well characterized. Our result is a key step towards the study of the different dynamics of solutions of the corresponding nonlinear Schrödinger and Klein-Gordon equations with energies above the energy of the ground state. Our restriction on the dimension is mainly due to the existence of resonances in dimension three and four.

This paper is a sequel to previous ones 38, 39, 41. We continue the study of the blowup problem for the nonlinear Schrödinger 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.

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