Compact Sobolev-Slobodeckij embeddings and positive solutions to fractional Laplacian equations

Abstract

In this work, we study the existence of a positive solution to an elliptic equation involving the fractional Laplacian (-Δ)s in ℝn, for n ≥ 2, such as (Formula Presented) Here, s ∈ (0, 1), q∈[2,2s∗) with 2s∗:=2nn-2s being the fractional critical Sobolev exponent, E(x), K(x), V(x) > 0: ℝn → ℝ are measurable functions which satisfy joint "vanishing at infinity"conditions in a measure-theoretic sense, and f (u) is a continuous function on ℝ of quasi-critical, super-q-linear growth with f (u) ≥ 0 if u ≥ 0. Besides, we study the existence of multiple positive solutions to an elliptic equation in ℝn such as (-Δ)su+E(x)u+V(x)uq-1=λK(x)ur-1, where 2 < r < q < ∞(both possibly (super-)critical), E(x), K(x), V(x) > 0: ℝn → ℝ are measurable functions satisfying joint integrability conditions, and λ > 0 is a parameter. To study (0.1)-(0.2), we first describe a family of general fractional Sobolev-Slobodeckij spaces Ms;q,p(ℝn) as well as their associated compact embedding results.