On the sharpness of a limiting case of the Sobolev imbedding theorem

Abstract

A refinement of the Sobolev imbedding theorem, due to Trudinger, is shown to be optimal in a natural sense.

Optimal domains for the kernel operator associated with Sobolev's inequality

“…We may choose the class of Lebesgue spaces. In this context, for inequality (1) the space L q (Ω) with q := np/(n − p) is the optimal range space given the fixed domain space L p (Ω), 1 < p < n. In the class of Orlicz spaces, the space L ϕ (Ω) with ϕ(t) = exp(t n ) − 1 and n := n/(n − 1) is the optimal range space for the fixed domain space L n (Ω) (see [11]). The optimal range problem within the class of Orlicz spaces has been studied by Cianchi [4].…”

An elementary proof of sharp Sobolev embeddings

“…In other words, neither of the target spaces can be replaced by an essentially smaller Orlicz space. This fact was observed by Hempel, Morris and Trudinger [9] for (1.2); a general result was later obtained by Cianchi [5]. However, both of the target spaces can be improved if we are willing to allow different function spaces than Orlicz spaces.…”

Eigenfunctions of the nonlinear equation Δu+νf(x,u) = 0 inR2

“…Theorems 1 and 2 are related to a question raised in Hempel-MorrisTrudinger [4]. Our approach in proving Theorem 1 is to look for critical points of the functional Φ{u) = \j$\Vu\ 2 on the surface S μ = [u e H&Q) I / Q .F(JC, w) = μ}.…”