Cases of equality and strict inequality in the extended Hardy–Littlewood inequalities

Abstract

Extended Hardy-Littlewood inequalities arewhere {u i } 1 i m are non-negative functions and {u * i } 1 i m denote their Schwarz symmetrization.In this paper, we determine appropriate conditions under which equality in ( * ) occurs if and only if {u i } 1 i m are Schwarz symmetric.

On the symmetry of minimizers in constrained quasi-linear problems

“…For results in this direction, under strict monotonicity assumptions of f such as f (|x|, s) > f (|y|, s), for all s ∈ R + and x, y ∈ Ω with |x| < |y|, we refer the reader to [16,Section 6] (see also [5]).…”

A sharp Gagliardo-Nirenberg inequality and its application to fractional problems with inhomogeneous nonlinearity

“…We shall see that Theorem 4.10 will play a crucial role to prove Theorem 1.7. We believe that Theorem 4.10 is of independent interest as Strichartz estimates is a celebrated tool to study short and long time behaviour of solution of (31). Theorem 4.10 is new and improves following theorem.…”

“…Theorem 4.11 (see Corollary 1 in [16]). Let N ≥ 2, N 2N −1 < s 1 ≤ s 2 ≤ 1 and ψ, ψ 0 , F are spherically symmetric in space and satisfying (31). Then…”

Some qualitative studies of the focusing inhomogeneous Gross–Pitaevskii equation

“…We now prove that ϕ is necessarily radial and radially decreasing. Indeed, denoting by ϕ * the Schwarz rearrangement of ϕ, it is well known that (see [22])ˆR Thus, from ∇ϕ * 2 L 2 ≤ ∇ϕ 2 L 2 , we infer that if ϕ is not radial, then S ω (ϕ * ) < S ω (ϕ) = d ω and K ω (ϕ * ) < K ω (ϕ) = 0, a contradiction. This prove that ϕ is radial and radially decreasing.…”