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The paper deals with the existence of normalized solutions to the systemfor any µ 1 , µ 2 , a 1 , a 2 > 0 and β < 0 prescribed. We present a new approach that is based on the introduction of a natural constraint associated to the problem. We also show that, as β → −∞, phase separation occurs for the solutions that we find. Our method can be adapted to scalar nonlinear Schrödinger equations with normalization constraint, and leads to alternative and simplified proofs to some results already available in the literature.

We consider the system of coupled elliptic equationsand study the existence of positive solutions satisfying the additional conditionAssuming that a 1 , a 2 , µ 1 , µ 2 are positive fixed quantities, we prove existence results for different ranges of the coupling parameter β > 0. The extension to systems with an arbitrary number of components is discussed, as well as the orbital stability of the corresponding standing waves for the related Schrödinger systems. PAT". This work has been carried out in the framework of the project NONLOCAL (ANR-14-CE25-0013), funded by the French National Research Agency (ANR). arXiv:1506.02262v1 [math.AP] 7 Jun 2015Lemma 4.9. There holds inf V J = inf V rad J.Proof. In order to prove the lemma we assume by contradiction that there exists (u, v) ∈ V such that (4.10) 0 < J(u, v) < inf V rad J.

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We prove the existence of infinitely many solutions λ 1 , λ 2 ∈ R, u, v ∈ H 1 (R 3 ), for the nonlinear Schrödinger systemwhere a, µ > 0 and β ≤ −µ are prescribed. Our solutions satisfy u = v so they do not come from a scalar equation. The proof is based on a new minimax argument, suited to deal with normalization conditions. 2010 Mathematics Subject Classification. 35J50, 35J15, 35J60.1 1 Indeed, in [16] the authors exploit a uniform-in-β Palais-Smale condition to derive the convergence of the whole minimax structure to a limit problem.2 In [6] we showed that P β is a C 1 manifold since this was enough for our purpose, the extra regularity is straightforward.

For a class of systems of semi-linear elliptic equations, including (Formula presented.), for p = 2 (variational-type interaction) or p = 1 (symmetric-type interaction), we prove that uniform L^∞ boundedness of the solutions implies uniform boundedness of their Lipschitz norm as β→+∞, lthat is, in the limit of strong competition. This extends known quasi-optimal regularity results and covers the optimal case for this class of problems. The proofs rest on monotonicity formulae of Alt–Caffarelli–Friedman and Almgren type in the variational setting, and on the Caffarelli–Jerison–Kenig almost monotonicity formula in the symmetric one

We study existence and phase separation, and the relation between these two aspects, of positive bound states for the nonlinear elliptic system (Formula presented.).This system arises when searching for solitary waves for the Gross–Pitaevskii equations. We focus on the case of simultaneous cooperation and competition, that is, we assume that there exist two pairs (i1,j1) and (i2,j2) such that i1≠j1, i2≠j2, βi1j1>0 and βi2,j2<0. Our first main results establishes the existence of solutions with at least m positive components for every m≤d; any such solution is a minimizer of the energy functional J restricted on a Nehari-type manifoldN. At a later stage, by means of level estimates on the constrained second differential of J on N, we show that, under some additional assumptions, any minimizer of J on N has all nontrivial components. In order to prove this second result, we analyse the phase separation phenomena which involve solutions of the system in a not completely competitive framework