Existence And Multiplicity Of Positive Solutions For Kirchhoff Type Problems With Singularity In High Dimensions

First,we consider the following Kirchhoff type problem with singularity and critical exponent where Ω is a bounded smooth domain in R4 and a,b>0,μ,λ>0,γ∈(0,1), 0≤β<3 is a constant.Now,we state our main results. Theorem 1.Assume a,b>0,0<γ<1,0≤β<3,and μ>0.Then there ex-ists λ*>0 such that problem (3)has at least one positive solution for all 0<λ<λ* Theorem 2.Assume a,b>0,0<γ<1/2+2λ<β<3 and μ>bS2.Then there exists λ**>0 such that problem(3) has at least two positive solutions for all 0<λ<λ**.Then we consider the following Kirchhoff type problem with singularity and subcritical exponent where Ω is a smooth bounded domain in RN (N≥4), a, b> 0,0<p< 2*-1 and 0<λ< 1 is a positive constant.f∈L2*/2*-1-p(Ω) and g∈L2*/2*-1+γ(Ω) are nonzero non-negative functions.Now, we state the main result.Theorem 3. Assume that a,b>0,0<p<2*-1,0<γ<1 and and are nonzero non-negative functions. Then problem (4) has at least one positive ground state solution u0 satisfying I(u0)<0.