Non-resonant states for Schrödinger–Poisson critical systems in high dimensions

Abstract

We prove a strong stability result with respect to the phase ω for Schrödinger-Poisson critical systems in the case of closed manifolds of dimensions n ≥ 6. As an application we prove the non-existence of solutions for Schrödinger-Poisson critical systems when ω 2 is large and n ≥ 6. These results are in contrast the 3, 4, 5-dimensional cases where resonant states do exist.Mathematics Subject Classification. 58J37, 35J47.

Blow-up for Schrödinger-Poisson critical systems in dimensions 4 and 5

“…Our results are in striking contrast with the lower dimensional case n = 3 and the higher dimensional cases n ≥ 6 where, by Hebey and Wei [12] and Thizy [21], no such sequences of solutions exist. The phenomenon we point out in this paper is very particular to dimensions 4 and 5.…”

“…We refer also to [11], where all the material needed for such an extension can be found). The result was extended to dimensions n ≥ 6 by Thizy [21], where it is proved that the same conclusions holds true (we assume in addition 16πq 2 > 1 if n = 6). We prove here that the remaining dimensions 4 and 5 are in striking contrast with these dimensions.…”

“…If we replace the pure power nonlinearity 2 − 1 by p − 1 in the right hand side of the first equation in (1.1), then the system behaves in a different manner. Assuming for instance that p ≤ 3, then the arguments in [21] give that on any closed Riemannian manifold (M n , g), n ≥ 3, the set of smooth positive solutions of the equation is bounded for all ω and even empty when ω 2 1 (only assuming that 16πq 2 > 1 if p = 3). An obstruction to get the existence of smooth positive solutions in this case is the loss of mountain-pass geometry pointed out in Ruiz and Siciliano [18].…”

Schrödinger-Poisson systems in $4$-dimensional closed manifolds

“…The existence of resonant states in 4 and 5-dimensions is investigated in Thizy [29] (we refer to Thizy [31] for the cases of dimensions n ≥ 6). The following theorem is proved in [29].…”

Electro-magneto-static study of the nonlinear Schrödinger equation coupled with Bopp-Podolsky electrodynamics in the Proca setting

“…When a = 0 we are back to the Schrödinger-Poisson system in Proca form as investigated in Hebey and Wei [30] and Thizy in the series of papers [46,47,48,49,50]. We assume from now on that we are in the static case of the system, for which ∂ t u ≡ 0, ∂ t A ≡ 0 and ∂ t ϕ ≡ 0, and we look for solutions with…”