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In this paper, we consider minimization problems related to the combined power-type nonlinear scalar field equations involving the Sobolev critical exponent in three space dimensions. In four and higher space dimensions, it is known that for any frequency and any power of the subcritical nonlinearity, there exists a ground state. In contrast to those cases, when the space dimension is three and the subcritical power is three or less, we can show that there exists a threshold frequency, above which no ground state exists, and below which the ground state exists. Furthermore, we prove the difference between two typical variational problems used to characterize the ground states.

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 [AIKN3], 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 Schr\"odinger 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., Comment: Revised version. See the first version for a proof of Proposition B.1. To appear in Calculus of Variations and Partial Differential Equations

We consider the nonlinear Schrodinger equations with combined type local interactions with energy critical growth, and we 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., Comment: To appear in Memoirs of AMS, 123 pages, 2 figures

We study the energy-critical focusing nonlinear Schr\"odinger equation with an energy- subcritical 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]., Comment: 2 figures, 48 pages

In this paper we show the existence of ground-state solutions for the energy-critical NLS perturbed with subcritical terms when the space dimension $d\geq4$. However in dimension three, we show that when the perturbation is small enough, then such solution does not exist. For the evolution equation, we show the existence of finite time blow up of solutions with radially symmetric data with energy below the one of the ground state., Comment: To appear in Advances in Differential Equations

We consider $L^{2}$-supercritical and $H^{1}$-subcritical focusing nonlinear Schr\"odinger equations. We introduce a subset $PW$ of $H^{1}(\mathbb{R}^{d})$ for $d\ge 1$, and investigate behavior of the solutions with initial data in this set. For this end, we divide $PW$ into two disjoint components $PW_{+}$ and $PW_{-}$. Then, it turns out that any solution starting from a datum in $PW_{+}$ behaves asymptotically free, and solution starting from a datum in $PW_{-}$ 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.