Multiplicity and asymptotics of standing waves for the energy critical half-wave

Abstract

In this paper, we consider the multiplicity and asymptotics of standing waves with prescribed mass R N u 2 = a 2 to the energy critical half-wavewhere N ≥ 2, a > 0, q ∈ 2, 2 + 2 N , 2 * = 2N N −1 and λ ∈ R appears as a Lagrange multiplier. We show that (0.1) admits a ground state u a and an excited state v a , which are characterised by a local minimizer and a mountain-pass critical point of the corresponding energy functional. Several asymptotic properties of {u a }, {v a } are obtained and it is worth pointing out that we get a precise description of {u a } as a → 0 + without needing any uniqueness condition on the related limit problem. The main contribution of this paper is to extend the main results in J. Bellazzini et al. [Math. Ann. 371 (2018), 707-740] from energy subcritical to energy critical case. Furthermore, these results can be extended to the general fractional nonlinear Schrödinger equation with Sobolev critical exponent, which generalize the work of H. J. Luo-Z. T. Zhang [Calc. Var. Partial Differ. Equ. 59 (2020)] from energy subcritical to energy critical case.

Multiplicity and asymptotics of positive normalized solutions for the fractional Sobolev critical Schr\"{o}dinger equations

“…Moreover, c 0 > 0 can (2) In the proof of Theorem 1.2, the main difficulty is to obtain the compactness of constrained Palais-Smale sequence at positive energy level. As pointed out in the very recent work [37],…”

“…In preparing this paper, we notice that in the very recent work [37], the question (Q) has been solved for N ⩾ 4s and s = 1 2 . Thus, it only need to consider the case 2s < N < 4s for the question (Q).…”

“…By using the ground state u + c and the Aubin-Talanti babbles to construct a good path, whose energy can be well controlled from above to ensure that it is smaller than the compact threshold, Luo-Yang-Yang [37] prove the existence of critical points of J µ | S (c) of Mountain Pass type for N ⩾ 4s, s = 1 2 and 2 < q < 2+ 4s N . Unlike [37] where a nonradial test function consisting of u + c and a bubble at infinity is used, here, inspired by Wei-Wu [48], we directly use the radial superposition of u + c and the Aubin-Talenti bubble. This test function seems to be more natural and it works for s ∈ (0, 1) in all dimensions N > 2s.…”

“…Remark 1.4. In the proof of Theorem 1.3, we follow some ideas from [37] to establish a refined upper bound of m + (c), see Lemma 5.2, which is the key ingredient to capture the precise description of ground state solutions of (1.2).…”