Existence Of Positive Solutions For Kirchhoff Type Problems With Hardy-Sobolev Critical Exponents And Singular Nonlinearities

We consider the Kirchhoff type problem with Hardy-Sobolev critical exponents and singular nonlinearities where Ω(?)R3 is a bounded domain with smooth boundary(?)Ω a>0,b≥ 0,0 ≥ s≤1.In this thesis,under the different conditions on f(x,u),we use the variational and perturbation methods to estimate the critical value and prove the local(PS)condition,then get the existence of positive solutions using the Mountain Pass Lem-ma and the strong maximum principle.There are our results:(1)When f(x,u)satisfies the following conditionsThen we can obtain the following theoremTheorem 1 Suppose a,b>0,0<b<A-2,s = 1,and f satisfies(f1),(f2),then problem(H)admits at least one positive solution.(where A be the Hardy-Sobolev constant)Theorem 2 Suppose a,b>0,s = 1,0<γ<1,0 ≤β<5/2,then there exists λ*>0 such that problem(H)has at least a positive solution for all 0<λ<λ*Theorem 3 Suppose a>0,0<b<A-2,s = 1,0<γ<5/6,5/3+ γ<β<5/2,then there exists λ**>0 such that problem(H)has at least two positive solutions for all 0<λ<λ**.(3)When f(x,u)= 0,one hasTheorem 4 Suppose a>0,b≥0,s = 1,when Ω(?)R3 is open,bounded,smooth and strictly starshaped with respect to 0,then problem(H)has no nontrivial solution.(4)When f(x,u)＝λu,one hasTheorem 5 Suppose a>0.b≥0.0≤s<1,0<λA<aλ1,b<As-2 then problem(H)admits at least one positive solution.(where As be the Hardy-Sobolev constant)...