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In this paper, we study the bifurcation problem from a line soliton for a stationary nonlinear Schrödinger equation on the product space R × T . We extend earlier results to a larger class of the nonlinearity in the equation. The salient point of our analysis relies on a lower bound of solution to the "auxiliary equation" and then on the application of the Crandall–Rabinowitz argument. [ABSTRACT FROM AUTHOR]

We consider the uniqueness of ground states for combined power-type nonlinear scalar field equations involving the Sobolev critical exponent at high frequencies. The uniqueness of ground states at high frequencies in five and higher dimensions has been proved in Akahori et al. (Calc. Var. Partial Differential Equations 58:32, 2019). Moreover, that in three dimensions can be obtained from the result of Coles and Gustafson (Publ. Res. Inst. Math. Sci. 56:647–699, 2020). On the other hand, the uniqueness in four dimensions has not been completely revealed. The aim in this paper is to prove the uniqueness in three and four dimensions in a unified way. Thus, we obtain a complete answer to the uniqueness problem for ground states at high frequencies. From the point of view of limiting profile of ground states at infinite frequency which is known to be the Aubin–Talenti function, the uniqueness problem in three and four dimensions is more difficult than that in higher dimensions. In this paper, we employ the fixed-point argument developed in Coles and Gustafson (Publ. Res. Inst. Math. Sci. 56:647–699, 2020). Since the application of the argument of Coles and Gustafson (Publ. Res. Inst. Math. Sci. 56:647–699, 2020) to four dimensions is by no means straightforward, we need to construct some estimates for the perturbed resolvents which fit the fixed-point argument (see Proposition 1.2 and (1.17)). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Our aim is to verify that the functional in the virial identity classifies the dynamics for nonlinear Schrödinger equations of local interactions. In particular, we give a condition under that there exist stable ground states. Our proof of this stability result is based on the ideas in Colin (Ann Inst H Pincaré 23:753–764, 2006) and Shatah (Math Phys 91:313–327, 1983). However, we emphasize that our argument does not use the strict convexity of the H ˙ 1 -norm of ground state with respect to ω : a key lemma is Lemma 4.8 below. Furthermore, we discuss the limiting profile of ground states (see Theorem 4.4). [ABSTRACT FROM AUTHOR]

We study the energy-critical focusing nonlinear Schrödinger 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 and Merle (Invent Math 166:645-675, ). [ABSTRACT FROM AUTHOR]

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 Akahori et al. (Global dynamics above the ground state energy for the combined power-type nonlinear Schrödinger equation with energy-critical growth at low frequencies, preprint), 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. [ABSTRACT FROM AUTHOR]