Existence And Multiplicity Of Solutions For Two Kinds Of Kirchhoff Type Equations With Critical Exponents

In this paper, we firstly study the existence and multiplicity of positive solutions for Kirchhoff type problem in R3 involving critical exponent by using the mountain pass theorem in critical point theory. Secondly, we study the existence of positive ground state solutions for Kirchhoff type problem involving critical exponent with Dirichlet boundary value condition.Firstly, we consider the following Kirchhoff type problem with critical exponent and h satisfies the following conditions: (h0) h ∈ L6/(5-q)(R3), h≥0 and h ≠ 0.Then we can obtain the following theorem. Theorem 1 Suppose a nd (ho) holds. Then there exists A*> 0 such that problem (3) has at least two different positive solutions for λ∈(0,A*).Secondly, we consider the following Kirchhoff type problem involving critical exponent with Dirichlet boundary value condition where Ω(?)R3 is a bounded smooth domain, α, b> 0, and f satisfies the following conditions:Now we state our main result.Theorem 2 Suppose a, b> 0, and (f1)-(f4) hold. Then problem (4) has at least a positive ground state solution.