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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]

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 (see Theorems 1.1 and 1.2). Furthermore, we prove the difference between two typical variational problems used to characterize the ground states (see Theorem 1.3). [ABSTRACT FROM AUTHOR]

We consider nonlinear scalar field equations involving the Sobolev-critical exponent at high frequencies ω. Since the limiting profile of the ground states as ω → ∞ is the Aubin–Talenti function and degenerate in a certain sense, from the point of view of perturbation methods, the nondegeneracy problem for the ground states at high frequencies is subtle. In addition, since the limiting profile (Aubin–Talenti function) fails to lie in L 2 (R d) for d = 3 , 4 , the nondegeneracy problem for d = 3 , 4 is more difficult than that for d ≥ 5 and an applicable methodology is not known. In this paper, we solve the nondegeneracy problem for d = 3 , 4 by modifying the arguments in Akahori et al. (2020) and Akahori et al. (2019). We also show that the linearized operator around the ground state has exactly one negative eigenvalue. [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]