Linear instability and nondegeneracy of ground state for combined power-type nonlinear scalar field equations with the Sobolev critical exponent and large frequency parameter

Abstract

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.

On some nonlinear Schrödinger equations in $\bbr^N$

“…where N ≥ 3, t > 0, λ > 0 and 2 < q < 2 * = 2N N−2 . Based on our recent study on the normalized solutions of the above equation in [31], we prove that (1) the above equation has two positive radial solutions for N = 3, 2 < q < 4 and t > 0 sufficiently large, which gives a rigorous proof of the numerical conjecture in [14]; (2) there exists t * q > 0 for 2 < q ≤ 4 such that the above equation has ground-states for t ≥ t * q in the case of 2 < q < 4 and for t > t * 4 in the case of q = 4 while, the above equation has no ground-states for 0 < t < t * q for all 2 < q ≤ 4, which, together with the well-known results on groundstates of the above equation, almost completely solve the existence of ground-states to the above equation, except for N = 3, q = 4 and t = t * 4 . Moreover, based on the almost completed study on ground-states to the above equation, we introduce a new argument to study the normalized solutions of the above equation to prove that there exists 0 < ta,q < +∞ for 2 < q < 2+ 4 N such that the above equation has no positive normalized solutions for t > ta,q with R N |u| 2 dx = a 2 , which, together with our recent study in [31], gives a completed answer to the open question proposed by Soave in [30].…”

“…If 3 ≤ N ≤ 6 and (N +2)/(N −2) < q < 2 * then Pucci and Serrin in [28] proved that (1.3) with p = 2 * has at most one positive radial solution. Recently, Akahori et al in [1,3,4] and Coles and Gustafson in [13] proved that the radial ground-state of (1.3) with p = 2 * is unique and nondegenerate for all small t > 0 when N ≥ 5 and q ∈ (2, 2 * ) or N = 3 and q ∈ (4, 2 * ); and for all large t > 0 when N ≥ 3 and 2 + 4/N < q < 2 * . However, the uniqueness of positive radial solutions seems not true for (1.3) with p = 2 * in general, since it is suggested in [14] by the numerical evidence that (1.3) with p = 2 * has two positive radial solutions for N = 3, 2 < q < 4 and t > 0 sufficiently large.…”

Global Existence and Singularity of the N-Body Problem with Strong Force

“…Proof of Theorem 4. Proof of (1). Suppose x(t) does not exist globally, by Painlevé's theorem, there exists σ > 0 so that lim t→σ min i =j r ij = 0.…”

“…The mass and the position of the i th particle is m i > 0 and x i ∈ R 3 , and letẋ i be its velocity. The potential is equal to (1) U…”

“…where r is the distance between two mass points. In the current paper, we would like to focus on the homogeneous potential N-body problem only, and the generalizations to quasi-homogenous N-body problem will be investigated in future work following ideas from [1] [2] [3]. The motion of the N-body is governed by the differential equation:…”