Monotonicity and rigidity of solutions to some elliptic systems with uniform limits

Abstract

In this paper we prove the validity of Gibbons' conjecture for a coupled competing Gross-Pitaevskii system. We also provide sharp a priori bounds, regularity results and additional Liouville-type theorems.

Saddle-shaped positive solutions for elliptic systems with bistable nonlinearity

“…that is, if and only if Λ > 1. Notice that, if (1.5) is violated, we have have non-existence of non-constant solutions [9,13], and hence (1.5) is sharp in this case. The proof of Theorem 1.1 consists in a 2 steps procedure.…”

“…Domain walls solutions satisfying asymptotic conditions (1.2) (u, v) → (1, 0) as x N → +∞, (u, v) → (0, 1) as x N → −∞, , in dimension N = 1 have been carefully studied in [2,4], where in particular it is shown the existence of such a solution for every Λ > 1 [4], and its uniqueness in the class of solutions with one monotone component [2]. In fact, uniqueness holds also without such assumption, and even in higher dimension [9]; precisely, in [9] it is shown that a solution to (1.1)-(1.2) (with the limits being uniform in x ′ ∈ R N −1 ) in R N with Λ > 1 is necessarily montone in both the components with respect to x N , and 1-dimensional. The assumption Λ > 1 is natural, since (1.1)-(1.2) has no solution at all when Λ ∈ (0, 1].…”

“…The assumption Λ > 1 is natural, since (1.1)-(1.2) has no solution at all when Λ ∈ (0, 1]. Indeed, it is proved that (1.1) has only constant solutions for both Λ ∈ (0, 1) [9], and Λ = 1 [9,13].…”

Phase transition in a Rabi coupled two-component Bose–Einstein condensate

“…1 The segregation case corresponds to Λ > 1 (1.4) and we will study the limits Λ → 1 and Λ → ∞. Let us point out that in the case ω = 0, the solution goes to (0, 1) and (1,0) at ±∞ and this problem has been analyzed in [3,10,19]. In [10], it is proved by the moving plane method that the solution is unique, u ′ > 0 and v ′ < 0.…”

“…Let us point out that in the case ω = 0, the solution goes to (0, 1) and (1,0) at ±∞ and this problem has been analyzed in [3,10,19]. In [10], it is proved by the moving plane method that the solution is unique, u ′ > 0 and v ′ < 0. The asymptotic behaviour for large Λ has been studied in [3]: the solution approaches the hyperbolic tangent in the half space while the inner solution is given by a simpler system analyzed in [6,7].…”