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.

We consider the combined power-type nonlinear Schrödinger equations with energy-critical growth, and study the solutions slightly above the ground state threshold at low frequencies, so that we obtain a so-called nine-set theory developed by Nakanishi and Schlag.

We consider combined power-type nonlinear scalar field equations with the Sobolev critical exponent. In [3], it was shown that if the frequency parameter is sufficiently small, then the positive ground state is nondegenerate and linearly unstable, together with an application to a study of global dynamics for nonlinear Schrödinger equations. In this paper, we prove the nondegeneracy and linear instability of the ground state frequency for sufficiently large frequency parameters. Moreover, we show that the derivative of the mass of ground state with respect to the frequency is negative.