We consider the existence of normalized solutions in H1(ℝN) × H1(ℝN) for systems of nonlinear Schr¨odinger equations, which appear in models for binary mixtures of ultracold quantum gases. Making a solitary wave ansatz, one is led to coupled systems of elliptic equations of the form

and we are looking for solutions satisfying

where a1> 0 and a2> 0 are prescribed. In the system, λ1 and λ2 are unknown and will appear as Lagrange multipliers. We treat the case of homogeneous nonlinearities, i.e. , with positive constants β, μi, pi, ri. The exponents are Sobolev subcritical but may be L2-supercritical. Our main result deals with the case in which in dimensions 2 ≤ N ≤ 4. We also consider the cases in which all of these numbers are less than 2 + 4/N or all are bigger than 2 + 4/N.

We introduce the notion of a perfect path for a monomial algebra. We classify indecomposable non-projective Gorenstein-projective modules over the given monomial algebra via perfect paths. We apply the classification to a quadratic monomial algebra and describe explicitly the stable category of its Gorenstein-projective modules.

We prove a reverse Faber–Krahn inequality for the Cheeger constant, stating that every convex body in ℝ2 has an affine image such that the product between its Cheeger constant and the square root of its area is not larger than the same quantity for the regular triangle. An analogous result holds for centrally symmetric convex bodies with the regular triangle replaced by the square. We also prove a Mahler-type inequality for the Cheeger constant, stating that every axisymmetric convex body in ℝ2 has a linear image such that the product between its Cheeger constant and the Cheeger constant of its polar body is not larger than the same quantity for the square.

The aim of this note is to study octahedrality in vector-valued Lipschitz-free Banach spaces on a metric space, under topological hypotheses on it, by analysing the weak-star strong diameter 2 property in Lipschitz function spaces. Also, we show an example that proves that our results are optimal and that octahedrality in vector-valued Lipschitz-free Banach spaces actually relies on the underlying metric space as well as on the Banach one.

For a C*-algebra A, determining the Cuntz semigroup Cu(A ⊗) in terms of Cu(A) is an important problem, which we approach from the point of view of semigroup tensor products in the category of abstract Cuntz semigroups by analysing the passage of significant properties from Cu(A) to Cu(A)⊗Cu Cu(). We describe the effect of the natural map Cu(A) → Cu(A)⊗Cu Cu() in the order of Cu(A), and show that if A has real rank 0 and no elementary subquotients, Cu(A)⊗Cu Cu() enjoys the corresponding property of having a dense set of (equivalence classes of) projections. In the simple, non-elementary, real rank 0 and stable rank 1 situation, our investigations lead us to identify almost unperforation for projections with the fact that tensoring with is inert at the level of the Cuntz semigroup.

We investigate the following nonlinear Neumann boundary-value problem with associated p(x)-Laplace-type operator

where the function φ(x, v) is of type |v|p(x)−2v with continuous function p: → (1,∞) and both f : Ω × ℝ → ℝ and g : ∂Ω × ℝ → ℝ satisfy a Carathéodory condition. We first show the existence of infinitely many weak solutions for the Neumann problems using the Fountain theorem with the Cerami condition but without the Ambrosetti and Rabinowitz condition. Next, we give a result on the existence of a sequence of weak solutions for problem (P) converging to 0 in L∞-norm by employing De Giorgi's iteration and the localization method under suitable conditions.

By invoking some new ideas, we characterize Sobolev spaces Wα,p(ℝn) with the smoothness order α ∊ (0, 2] and p ∊ (max{1, 2n/(2α + n)},∞), via the Lusin area function and the Littlewood–Paley g*λ-function in terms of centred ball averages. We also show that the assumption p ∊ (max{1, 2n/(2α + n)},∞) is nearly sharp in the sense that these characterizations are no longer true when p ∊ (1, max{1, 2n/(2α + n)}). These characterizations provide a possible new way to introduce Sobolev spaces with smoothness order in (1, 2] on metric measure spaces.

Using the Leray–Schauder degree, we study the existence of solutions for the following periodic differential equation with relativistic acceleration and singular nonlinearity:

where μ > 1 and the weight h: [0, T] → ℝ is a continuous sign-changing function. There are no a priori estimates on the set of positive solutions (a condition used in general to apply the Leray–Schauder degree), and we prove that no solution of the equation appears on the boundary of an unbounded open set during the deformation to an autonomous problem.

We study a quasilinear degenerately elliptic system involving the operator curl. The leading-order term of the associated energy functional is a q-power of curl of the unknowns. It is interesting to us that the structure of the lower-order terms will play an important role in both the existence and regularity of the solutions. When the lower-order part of the energy functional is convex we obtain weak solutions by minimizing the functional in some suitable spaces of vector fields. When it is concave we obtain critical points of the truncated functional, which are weak solutions of a nonlinear eigenvalue problem. We also examine the interior Hölder regularity of the weak solutions, which indicates that the weak solution is composed of a ‘good’ divergence-free part and a ‘bad’ part in gradient form. The analysis involves local higher integrability and local Hölder gradient estimates for a translated q′-Laplace equation and a translated quasilinear equation with p-growth.

Functions of bounded deformation (BD) arise naturally in the study of fracture and damage in a geometrically linear context. They are related to functions of bounded variation (BV), but are less well understood. We discuss here the relation to BV under additional regularity assumptions, which may require the regular part of the strain to have higher integrability or the jump set to have finite area or the Cantor part to vanish. On the positive side, we prove that BD functions that are piecewise affine on a Caccioppoli partition are in GSBV, and we prove that SBDp functions are approximately continuous -almost everywhere away from the jump set. On the negative side, we construct a function that is BD but not in BV and has distributional strain consisting only of a jump part, and one that has a distributional strain consisting of only a Cantor part.

The paper studies large time behaviour of solutions to the Keller–Segel system with quadratic degradation in a liquid environment, as given by

under Neumann boundary conditions in a bounded domain Ω ⊂ ℝn, where n ≥ 1 is arbitrary. It is shown that whenever U : Ω × (0,∞) → ℝn is a bounded and sufficiently regular solenoidal vector field any non-trivial global bounded solution of (⋆) approaches the trivial equilibrium at a rate that, with respect to the norm in either of the spaces L1(Ω) and L∞(Ω), can be controlled from above and below by appropriate multiples of 1/(t + 1). This underlines that, even up to this quantitative level of accuracy, the large time behaviour in (⋆) is essentially independent not only of the particular fluid flow, but also of any effect originating from chemotactic cross-diffusion. The latter is in contrast to the corresponding Cauchy problem, for which known results show that in the n = 2 case the presence of chemotaxis can significantly enhance biomixing by reducing the respective spatial L1 norms of solutions.

We describe the structure of the Ricci tensor on a locally homogeneous Lorentzian gradient Ricci soliton. In the non-steady case, we show that the soliton is rigid in dimensions 3 and 4. In the steady case we give a complete classification in dimension 3.

We consider the parabolic one-dimensional Allen–Cahn equation

The steady state connects, as a ‘transition layer’, the stable phases –1 and +1. We construct a solution u with any given number k of transition layers between –1 and +1. Mainly they consist of k time-travelling copies of w, with each interface diverging as t → –∞. More precisely, we find

where the functions ξj (t) satisfy a first-order Toda-type system. They are given by

for certain explicit constants γjk.

By considering the norm of a locally univalent function given by

we obtain such norm estimates for an operator of functions involving a convolution structure of convex univalent functions with a subclass of convex functions (defined by subordination). We also obtain some inequalities concerning this norm for functions under a certain fractional integral operator. Some implications of our results are briefly pointed out in the concluding section.

We study the Lane–Emden equation in strips.

In this paper the existence and uniqueness of weak and strong solutions for a non-autonomous non-local reaction–diffusion equation is proved. Furthermore, the existence of minimal pullback attractors in the L2-norm in the frameworks of universes of fixed bounded sets and those given by a tempered growth condition is established, along with some relationships between them. Finally, we prove the existence of minimal pullback attractors in the H1-norm and study relationships among these new families and those given previously in the L2 context. We also present new results in the autonomous framework that ensure the existence of global compact attractors as a particular case.

We consider the nonlinear Dirac equation

The potential function V satisfies the conditions that the essential spectrum of the Dirac operator is and this Dirac operator has infinitely many eigenvalues in (−1, 1) accumulating at 1. This potential function V may change sign in ℝ3 and contains the classical Coulomb potential V (x) = −γ/|x| with γ > 0 as a special case. The nonlinearity F satisfies the resonance-type condition lim. Under some additional conditions on V and F, we prove that this equation has infinitely many solutions.

We study some Hardy-type inequalities involving a general norm in ℝn and an anisotropic distance function to the boundary. The case of the optimality of the constants is also addressed.

This paper is devoted to the study of the long wave approximation for water waves under the influence of the gravity and a Coriolis forcing. We start by deriving a generalization of the Boussinesq equations in one (spatial) dimension and we rigorously justify them as an asymptotic model of water wave equations. These new Boussinesq equations are not the classical Boussinesq equations: a new term due to the vorticity and the Coriolis forcing appears that cannot be neglected. We study the Boussinesq regime and derive and fully justify different asymptotic models when the bottom is flat: a linear equation linked to the Klein–Gordon equation admitting the so-called Poincaré waves; the Ostrovsky equation, which is a generalization of the Korteweg–de Vries (KdV) equation in the presence of a Coriolis forcing, when the rotation is weak; and the KdV equation when the rotation is very weak. Therefore, this work provides the first mathematical justification of the Ostrovsky equation. Finally, we derive a generalization of the Green–Naghdi equations in one spatial dimension for small topography variations and we show that this model is consistent with the water wave equations.

Let f : ℕ → ℂ be a bounded multiplicative function. Let a be a fixed non-zero integer (say a = 1). Then f is well distributed on the progression n ≡ a (mod q) ⊂ {1,…, X}, for almost all primes q ∈ [Q, 2Q], for Q as large as X1/2+1/78−o(1).