Existence and phase separation of entire solutions to a pure critical competitive elliptic system

Abstract

We establish the existence of a positive fully nontrivial solution (u, v) to the weakly coupled elliptic systemwhere N ≥ 4, 2 * := 2N N−2 is the critical Sobolev exponent, α, β ∈ (1, 2], α + β = 2 * , µ 1 , µ 2 > 0, and λ < 0. We show that these solutions exhibit phase separation as λ → −∞, and we give a precise description of their limit domains.If µ 1 = µ 2 and α = β, we prove that the system has infinitely many fully nontrivial solutions, which are not conformally equivalent.Pitaevskii equations. This type of systems arises, e.g., in the Hartree-Fock theory for double condensates, that is, Bose-Einstein condensates of two different hyperfine states which overlap in space; see [9]. The sign of µ i reflects the interaction of the particles within each single state. If µ i is positive, this interaction is attractive. The sign of λ, on the other hand, reflects the interaction of particles in different states. This interaction is attractive if λ > 0 and it is repulsive if λ < 0. If the condensates Date: September 29, 2018. M. Clapp was partially supported by CONACYT grant 237661 (Mexico) and UNAM-DGAPA-PAPIIT grant IN104315 (Mexico). A. Pistoia was partially supported by Fondi di Ateneo "Sapienza" Universitá di Roma (Italy).1 2 * 2 for all λ > 0 if N ≥ 5 and for a wide range of λ > 0 if N = 4; see [5,6]. Peng, Peng and Wang [20] studied the system for µ 1 = µ 2 = 1, λ = 1 2 * and different values of α and β, and they obtained uniqueness and nondegeneracy results for positive synchronized solutions. Guo, Li and Wei studied the critical system (1.1) in dimension N = 3 for λ < 0 and they established the existence of positive solutions with k peaks for k sufficiently large in [12]. In [10,11] Gladiali, Grossi and Troestler obtained radial and nonradial solutions to some critical systems using bifurcation methods.Here we focus our attention to the competitive case, i.e., to λ < 0. In this case, the system (1.1) does not have a least energy fully nontrivial solution; see Proposition 2.2 below. This behavior showcases the lack of compactness of the variational functional, which comes from the fact that system is invariant under

A simple variational approach to weakly coupled competitive elliptic systems

“…One of them is positive and has least energy among all Γ-invariant fully nontrivial solutions. Theorem 1.2 extends some earlier results obtained in [5,6] for a system of two equations; see also [9]. Existence and multiplicity results for the purely critical system in a bounded domain may be found in [5,13,14].…”

“…Proof. Following the argument given in [6,Proposition 2.2] for M = 2 one can easily prove this statement.…”

“…To obtain multiple solutions to the system (1.3) we consider a conformal action on R N , as in [6,8].…”

“…for any γ ∈ Γ and w ∈ D 1,2 (R N ); see [6,Section 3]. We shall say that w is Γ-invariant if γw = w for all γ ∈ Γ, and that (u 1 , .…”

Symmetry and monotonicity properties of singular solutions to some cooperative semilinear elliptic systems involving critical nonlinearities

“…Papers on existence or qualitative properties of solutions to systems with critical growth in R n are very few, due to the lack of compactness given by the Talenti bubbles and the difficulties arising from the lack of good variational methods. We refer the reader to [9,13,24,25,26,35] for this kind of systems. The starting point of the second part of the paper is the study of qualitative properties of singular solutions to the following m × m system of equations…”