Sobolev type inequalities on Riemannian manifolds

A characterization related to Schrödinger equations on Riemannian manifolds

“…Theorem A. [PRS08, Corollary 2.17], [AX10]. Let (M, g) be an N−dimensional complete Riemannian manifold.…”

“…In order to give the precise statement of our result, let us denote by (M, g) a N-dimensional (N ≥ 3), complete, non-compact Riemannian manifold with asymptotically non-negative Ricci curvature with a base pointx 0 ∈ M, i.e., (C) Ric (M,g) (x) ≥ −(N − 1)H(d g (x 0 , x)), for all x ∈ M, where H ∈ C 1 ([0, ∞)) is a nonnegative bounded function satisfying ∞ 0 tH(t)dt = b 0 < +∞, (here and in the sequel d g is the distance function associated to the Riemannian metric g). For an overview on such property see [AX10,PRS08]. Let x 0 ∈ M be a fixed point, α : M → R + \ {0} a bounded function and f : R + → R + a continuous function with f (0) = 0 such that there exist two constants C > 0 and q ∈ (1, 2 ⋆ ) (being 2 ⋆ the Sobolev critical exponent) such that f (ξ) ≤ k 1 + ξ q−1 for all ξ ≥ 0.…”

Twoweak solutions for a singular (p,q)-Laplacian problem

“…This is the best constant (see [2]) of the Sobolev embedding theorem (see [33,Proposition B.7]). By Holder's inequality…”

The Caffarelli-Kohn-Nirenberg Inequalities on Metric Measure Spaces

“…In [1,2,9,10,16,30,31], the authors consider the study of Riemannian manifolds with non-negative Ricci curvature supporting some of the particular classes of CKN. In particular, in [1,2,9,30,31], the authors obtain some metric and topological rigidity results.…”