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Abstract We study ground states of two-component Bose–Einstein condensates (BEC) with trapping potentials in R 2 , where the intraspecies interaction (− a 1 , − a 2) and the interspecies interaction − β are both attractive, i. e , a 1 , a 2 and β are all positive. The existence and non-existence of ground states are classified completely by investigating equivalently the associated L 2 -critical constraint variational problem. The uniqueness and symmetry-breaking of ground states are also analyzed under different types of trapping potentials as β ↗ β ⁎ = a ⁎ + (a ⁎ − a 1) (a ⁎ − a 2) , where 0 < a i < a ⁎ : = ‖ w ‖ 2 2 (i = 1 , 2) is fixed and w is the unique positive solution of Δ w − w + w 3 = 0 in R 2. The semi-trivial limit behavior of ground states is tackled in the companion paper [12]. [ABSTRACT FROM AUTHOR]

In this paper we study a diffuse interface generalized antiferromagnetic model. The functional describing the model contains a Modica–Mortola type local term and a nonlocal generalized antiferromagnetic term in competition. The competition between the two terms results in a frustrated system which is believed to lead to the emergence of a wide variety of patterns. The sharp interface limit of our model is considered in [23] and in [10]. In the discrete setting it has been previously studied in [16,17,19]. The model contains two parameters: τ and ε. The parameter τ represents the relative strength of the local term with respect to the nonlocal one, while the parameter ε describes the transition scale in the Modica–Mortola type term. If τ < 0 one has that the only minimizers of the functional are constant functions with values in { 0 , 1 }. In any dimension d ≥ 1 for small but positive τ and ε , it is conjectured that the minimizers are non-constant one-dimensional periodic functions. In this paper we are able to prove such a characterization of the minimizers, thus showing also the symmetry breaking in any dimension d > 1. [ABSTRACT FROM AUTHOR]

In this paper, we will consider the fractional Caffarelli-Kohn-Nirenberg inequality Λ (∫ R n | u (x) | p | x | β p d x) 2 p ≤ ∫ R n ∫ R n (u (x) − u (y)) 2 | x − y | n + 2 γ | x | α | y | α d y d x where γ ∈ (0 , 1) , n ≥ 2 , and α , β ∈ R satisfy α ≤ β ≤ α + γ , − 2 γ < α < n − 2 γ 2 , and the exponent p is chosen to be p = 2 n n − 2 γ + 2 (β − α) , such that the inequality is invariant under scaling. We first study the existence and nonexistence of extremal solutions. Our next goal is to show some results on the symmetry and symmetry breaking region for the minimizers; these suggest the existence of a Felli-Schneider type curve separating both regions but, surprisingly, we find a novel behavior as α → − 2 γ. The main idea in the proofs, as in the classical case, is to reformulate the fractional Caffarelli-Kohn-Nirenberg inequality in cylindrical variables. Then, in order to find the radially symmetric solutions we need to solve a non-local ODE. For this equation we also get uniqueness of minimizers in the radial symmetry class; indeed, we show that the unique continuation argument of Frank-Lenzmann (Acta'13) can be applied to more general operators with good spectral properties. We provide, in addition, a completely new proof of non-degeneracy which works for all critical points. It is based on the variation of constants approach and the non-local Wronskian of Ao-Chan-DelaTorre-Fontelos-González-Wei (Duke'19). [ABSTRACT FROM AUTHOR]

Abstract: The correlation term, introduced in [13] to describe the interaction between very far apart vortices, governs symmetry-breaking for the Ginzburg–Landau equation in or bounded domains. It is a homogeneous function of degree , and then for -symmetric vortex configurations can be expressed in terms of the so-called correlation coefficient. Ovchinnikov and Sigal [13] have computed it in few cases and conjectured its value to be an integer multiple of . We will disprove this conjecture by showing that the correlation coefficient always vanishes, and will discuss some of its consequences. [Copyright &y& Elsevier]

Abstract: We consider the ground state solutions of the Lane–Emden system with Hénon-type weights , in the unit ball B of with Dirichlet boundary conditions, where , , and . We show that such ground state solutions u, v always have definite sign in B and exhibit a foliated Schwarz symmetry with respect to a unit vector of . We also give precise conditions on the parameters α, β, p and q under which the ground state solutions are not radially symmetric. [Copyright &y& Elsevier]

Abstract: We prove a new inequality which improves on the classical Hardy inequality in the sense that a nonlinear integral quantity with super-quadratic growth, which is computed with respect to an inverse square weight, is controlled by the energy. This inequality differs from standard logarithmic Sobolev inequalities in the sense that the measure is neither Lebesgue''s measure nor a probability measure. All terms are scale invariant. After an Emden–Fowler transformation, the inequality can be rewritten as an optimal inequality of logarithmic Sobolev type on the cylinder. Explicit expressions of the sharp constant, as well as minimizers, are established in the radial case. However, when no symmetry is imposed, the sharp constants are not achieved by radial functions, in some range of the parameters. [Copyright &y& Elsevier]