We study positive solutions to the fractional semi-linear elliptic equation $${\left( { - \Delta } \right)^\sigma }u = K\left( x \right){u^{{{n + 2\sigma \over {n }}\,}}\,\,\,{\rm{in}}\,{B_2}\backslash \left\{ 0 \right\}$$ with an isolated singularity at origin, where K is a function on B2, punctured ball B2 {0} ⊂ ℝn n ⩾ 2, σ ∈ (0, 1), and (−Δ)σ Laplacian. In lower dimensions, we show that for ...

The carré du champ method is a powerful technique for proving interpolation inequalities with explicit constants in presence of non-trivial metric on manifold. applies to some classical Gagliardo-Nirenberg-Sobolev the sphere, optimal constants. Very nonlinear regimes close critical Sobolev exponent can be covered using parabolic flows porous medium or fast diffusion type. Considering power law ...

In the present paper we deal with a quasilinear elliptic equation involvinga critical Sobolev exponent on non-compact Finsler manifolds, i.e. Randers spaces. Under very general assumptions perturbation prove existence of non-trivial solution. The approach is based direct methods calculus variations. One key step to that energy functional associated problem weakly lower semicontinuous small ball...

It is well known that Sobolev embeddings can be refined in the presence of symmetries. Hebey and Vaugon (1997) studied this phenomena in the context of an arbitrary Riemannian manifold M and a compact group of isometries G. They showed that the limit Sobolev exponent increases if there are no points in M with discrete orbits under the action of G. In the paper, the situation where M contains po...