The Properties Of Solutions Of The Fractional Laplace With Hardy Potential

The fractional Laplace is a non-local elliptic operator.It is widely used in many fields such as finance,medicine,physics,chemistry,hydrology and so on.Firstly,we obtain the existence and summbility of solutions to the following fractional equation with the Hardy potential and low order term（?）where s∈（0,1）,r>1,N>2s,Ω（?）R^N is a bounded domain with Lipschitz boundary such that 0 ∈Ω.Let f∈Lm（Ω）be a non-negative function.This thesis show that equation（0.1）has weak solution u ∈ H₀^s（Ω）∩L^∞（Ω）if m∈[1,1+1/p];equation（0.1）has finite energy solution if m ∈（1+1/p,N（p+1）2Ps];equation（0.1）has solution if m>N（P+1）2ps.Secondly,we consider the existence of solutions to the fractional elliptic equation with singular term（?）where Ω（?）R^N is a bounded domain with Lipschitz boundary.H（x,u）:Ω×[0,μ）→R is a Caratheodory function satisfying suitable hypotheses.Let f∈L¹（Ω）be a non-negative function.This thesis show that equation（0.2）has weak solution u∈H₀^s（Ω）∩L^∞).