Existence of Positive Ground State Solutions for Choquard Systems

We study the existence of positive ground state solution for Choquard systems. In the autonomous case, we prove the existence of at least one positive ground state solution by the Pohozaev manifold method and symmetric-decreasing rearrangement arguments. Moreover, we show that each positive ground state solution is radial symmetric. While, in the nonautonomous case, a positive ground state solution is obtained by using a monotonicity trick and a global compactness lemma. We remark that, under our assumptions of the nonlinearity {W_{u}}, the search of ground state solutions cannot be reduced to the study of critical points of a functional restricted to a Nehari manifold.

Symmetric radial decreasing rearrangement can increase the fractional Gagliardo norm in domains

We show that the symmetric radial decreasing rearrangement can increase the fractional Gagliardo semi-norm in domains.

Variable exponents and grand Lebesgue spaces: Some optimal results

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 [Formula: see text]. 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 Lp*(0))(Ω). This inclusion fails to be true if we impose a slower growth as [Formula: see text] at zero. Some other results are discussed.

An Odd Rearrangement ofL1(Rn)

We introduce an odd rearrangementf*defined byπ(f)(x)=f*(x)=sgn(x1)f*(νn|x|n),x∈Rn, wheref*is a decreasing rearrangement of the measurable functionf. With the help of this odd rearrangement, we show that for eachf∈L1(Rn), there exists ag∈H1(Rn)such thatdf=dg, wheredfis an distribution function off. Moreover, we study the boundedness of singular integral operators when they are restricted to odd rearrangement ofL1(Rn), and we give some results on Hilbert transform.

Estimates for Convex Integral Means of Harmonic Functions

We prove that if f is an integrable function on the unit sphere S in ℝn, g is its symmetric decreasing rearrangement and u, v are the harmonic extensions of f, g in the unit ball , then v has larger convex integral means over each sphere rS, 0 < r < 1, than u has. We also prove that if u is harmonic in with |u| < 1 and u(0) = 0, then the convex integral mean of u on each sphere rS is dominated by that of U, which is the harmonic function with boundary values 1 on the right hemisphere and −1 on the left one.