For the positive solutions of the Gross-Pitaevskii systemwe prove that L ∞ -boundedness implies C 0,α -boundedness, uniformly as β → +∞, for every α ∈ (0, 1). Moreover we prove that the limiting profile, as β → +∞, is Lipschitz continuous. The proof relies upon the blow-up technique and the monotonicity formulae by Almgren and Alt-Caffarelli-Friedman. This system arises in the Hartree-Fock approximation theory for binary mixtures of Bose-Einstein condensates in different hyperfine states. Extensions to systems with k > 2 densities are given.MSC : 35B40, 35B45, 35J55.

We deal with a class of Lipschitz vector functions U = (u1, . . . , u h ) whose components are non negative, disjointly supported and verify an elliptic equation on each support. Under a weak formulation of a reflection law, related to the Pohozaev identity, we prove that the nodal set is a collection of C 1,α hyper-surfaces (for every 0 < α < 1), up to a residual set with small Hausdorff dimension. This result applies to the asymptotic limits of reaction-diffusion systems with strong competition interactions, to optimal partition problems involving eigenvalues, as well as to segregated standing waves for Bose-Einstein condensates in multiple hyperfine spin states.

We consider a system of the form −ε 2 ∆u u) in an open domain Ω of R N , with Dirichlet conditions at the boundary (if any). We suppose that f and g are power-type non-linearities, having superlinear and subcritical growth at infinity. We prove the existence of positive solutions u ε and v ε which concentrate, as ε → 0, at a prescribed finite number of local minimum points of V (x), possibly degenerate.

Given ρ > 0, we study the elliptic problem find (U, λ) ∈ H 1 0 (B1) × R such thatthe unitary ball and p is Sobolev-subcritical. Such problem arises in the search of solitary wave solutions for nonlinear Schrödinger equations (NLS) with power nonlinearity on bounded domains. Necessary and sufficient conditions (about ρ, N and p) are provided for the existence of solutions. Moreover, we show that standing waves associated to least energy solutions are orbitally stable for every ρ (in the existence range) when p is L 2 -critical and subcritical, i.e. 1 < p ≤ 1 + 4/N , while they are stable for almost every ρ in the L 2 -supercritical regime 1 + 4/N < p < 2 * − 1. The proofs are obtained in connection with the study of a variational problem with two constraints, of independent interest: to maximize the L p+1 -norm among functions having prescribed L 2 and H 1 0 -norm. arXiv:1307.3981v1 [math.AP]

Abstract. Consider a Hamiltonian elliptic system of type    −∆u = Hv (u, v) in Ω −∆v = Hu (u, v) in Ω u, v = 0 on ∂Ω where H is a power-type nonlinearity, for instancehaving subcritical growth, and Ω is a bounded domain of R N , N ≥ 1. The aim of this paper is to give an overview of the several variational frameworks that can be used to treat such a system. Within each approach, we address existence of solutions, and in particular of ground state solutions. Some of the available frameworks are more adequate to derive certain qualitative properties; we illustrate this in the second half of this survey, where we also review some of the most recent literature dealing mainly with symmetry, concentration, and multiplicity results. This paper contains some original results as well as new proofs and approaches to known facts.