Our main purpose in this article is to establish a Gagliardo-Nirenberg type inequality in the critical Sobolev-Morrey space H M the growth order r 1− 1 p as r → ∞. The inequality (GN) implies that the growth order as r → ∞ is linear, which might look worse compared to the case of the critical Sobolev space. Communicated by Hans Triebel. Y. Sawano ( ) J Fourier Anal ApplHowever, we investigate the optimality of the growth order and prove that this linear order is best-possible. Furthermore, as several applications of the inequality (GN), we shall obtain a Trudinger-Moser type inequality and a Brézis-Gallouët-Wainger type inequality in the critical Sobolev-Morrey space.

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Let 1 ≤ p ≤ ∞. We show that a function u ∈ C(R N ) is a viscosity solution to the normalized p-Laplace equation n p u(x) = 0 if and only if the asymptotic formulaholds as ε → 0 in the viscosity sense. Here,with respect to λ ∈ R. This kind of asymptotic mean value property (AMVP) extends to the case p = 1 previous (AMVP)'s obtained when μ p (ε, u)(x) is replaced by other kinds of mean values. The natural definition of μ p (ε, u)(x) makes sure that this is a monotonic and continuous (in the appropriate topology) functional of u. These two properties help to establish a fairly general proof of (AMVP), that can also be extended to the (normalized) parabolic p-Laplace equation.

Let n 3 and Ω be a C 1 bounded domain in R n with 0 ∈ ∂Ω. Suppose ∂Ω is C 2 at 0 and the mean curvature of ∂Ω at 0 is negative, we prove the existence of positive solutions for the equation:where λ > 0, 0 < s < 2, 2 * (s) = 2(n−s) n−2 and n 4. For n = 3, the existence result holds for 0 < s < 1. Under the same assumption of the domain Ω, for p 2 * (s) − 1, we also prove the existence of a positive solution for the following equation:where λ > 0 and 1 p < n n − 2 .

In this article, we establish the weighted Trudinger–Moser inequality of the scaling invariant form including its best constant and prove the existence of a maximizer for the associated variational problem. The non-singular case was treated by Adachi and Tanaka (1999) [1] and the existence of a maximizer is a new result even for the non-singular case. We also discuss the relation between the best constants of the weighted Trudinger–Moser inequality and the Caffarelli–Kohn–Nirenberg inequality in the asymptotic sense.