Communications in Contemporary Mathematics

We study the existence and regularity of the solution to the multivalued equation -Δ_Φu ∈ ∂j(u) + λh in Ω, where Ω ⊂ R^N is a bounded smooth domain, Φ is an N-function, Δ_Φ is the corresponding Φ-Laplacian, λ > 0 is a parameter, h is a measurable function, and j is a continuous function with critical growth where ∂j(u) denotes its subdifferential. We apply the Ekeland Variational Principle to an associated locally Lipschitz energy functional. A major point in our study is that in order to deal with the obtained Ekeland sequence we developed a generalized version for the framework of Orlicz–Sobolev spaces of a well-known Brézis–Lieb lemma which was employed together with a variant of the Lions concentration-compactness theory to get a solution of the equation.

We study the following boundary value problem with a concave–convex nonlinearity:

We prove in this paper that the minimizer of Lennard–Jones energy per particle among Bravais lattices is a triangular lattice, i.e. composed of equilateral triangles, in ℝ² for large density of points, while it is false for sufficiently small density. We show some characterization results for the global minimizer of this energy and finally we also prove that the minimizer of the Thomas–Fermi energy per particle in ℝ² among Bravais lattices with fixed density is triangular.

We present results expressing conditions for the existence of meromorphic first integrals for Pfaff systems of arbitrary codimension on complex manifolds. Some of the results presented improve previous ones due to Jouanolou and Ghys. We also present an enumerative result counting the number of hypersurfaces invariant by a projective holomorphic foliation with split tangent sheaf.

The purpose of this paper is to illustrate the I-method by studying low-regularity solutions of the nonlinear Schrödinger equation in two space dimensions. By applying this method, together with the interaction Morawetz estimate, (see [J. Colliander, M. Grillakis and N. Tzirakis, Tensor products and correlation estimates with applications to nonlinear Schrödinger equations, Commun. Pure Appl. Math.62 (2009) 920–968; F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. École Norm. Sup.42 (2009) 261–290]), we establish global well-posedness and scattering for low-regularity solutions of the equation iu_t + Δu = λ₁|u|^p₁ u + λ₂|u|^p₂ u under certain assumptions on parameters. This is the first result of this type for an equation which is not scale-invariant. In the first step, we establish global well-posedness and scattering for low regularity solutions of the equation iu_t + Δu = |u|^p u, for a suitable range of the exponent p extending the result of Colliander, Grillakis and Tzirakis [Commun. Pure Appl. Math.62 (2009) 920–968].

This paper is concerned with the stability analysis of stationary solutions of the Cauchy–Dirichlet problem for some semilinear heat equation with concave nonlinearity. The instability of sign-changing solutions is verified under some variational assumption. Moreover, the exponential stability of the positive stationary solution at an optimal rate is proved by exploiting a super–subsolution method as well as the parabolic regularity theory. The base of our analysis relies on the linearization of the equation at each stationary solution and spectral analysis of the corresponding linearized operator. The main difficulties reside in the singularity of the linearized operator due to the concave nonlinearity.

Let Ω be a bounded domain of ℝ^N, and Q = Ω × (0, T). We consider problems of the type

In this paper, we provide necessary and sufficient conditions for a cleft right H-comodule algebra (A, ϱ_A) over a Hopf quasigroup H to be isomorphic as an algebra to the crossed product A_H♯_σAHH, where A_H is the coinvariants subalgebra of A and σ_AH is a morphism between H ⊗ H and A_H. As a consequence, we obtain the corresponding version in the nonassociative setting of the result given by Blattner, Cohen and Montgomery for projections of Hopf algebras with coalgebra splitting. Concrete examples satisfying the obtained conditions are provided.

We introduce the notions of almost h-slant Riemannian maps, almost h-semi-invariant Riemannian maps, and almost h-semi-slant Riemannian maps from Riemannian manifolds to almost quaternionic Hermitian manifolds. We investigate the harmonicity of such maps and the geometry of distributions. We also find the conditions for such maps to be totally geodesic, relate the notion of pseudo-horizontally weakly conformal maps to those notions, and give some examples of such maps.

Consider p : Ω → [1, +∞[, a measurable bounded function on a bounded set Ø with decreasing rearrangement p_* : [0, |Ω|] → [1, +∞[. We construct a rearrangement invariant space with variable exponent p_* denoted by . According to the growth of p_*, we compare this space to the Lebesgue spaces or grand Lebesgue spaces. In particular, if p_*(⋅) satisfies the log-Hölder continuity at zero, then it is contained in the grand Lebesgue space L^p_*(0))(Ω). This inclusion fails to be true if we impose a slower growth as at zero. Some other results are discussed.

In this paper, we consider a perturbation of the Ricci solitons equation proposed in [J.-P. Bourguignon, Ricci curvature and Einstein metrics, in Global Differential Geometry and Global Analysis, Lecture Notes in Mathematics, Vol. 838 (Springer, Berlin, 1981), pp. 42–63] and studied in [H.-D. Cao, Geometry of Ricci solitons, Chinese Ann. Math. Ser. B27(2) (2006) 121–142] and we classify noncompact gradient shrinkers with bounded non-negative sectional curvature.

We consider a parametric nonlinear Dirichlet problem driven by the p-Laplacian, with a singular term and a p-superlinear perturbation, which need not satisfy the usual Ambrosetti–Rabinowitz condition. Using variational methods together with truncation techniques, we prove a bifurcation-type theorem describing the behavior of the set of positive solutions as the parameter varies.

We present a formula for vector-valued modular forms, expressing the value of the Hilbert-polynomial of the module of holomorphic forms evaluated at specific arguments in terms of traces of representation matrices, restricting the weight distribution of the free generators.