Existence Of Solutions For Kirchhoff Equations With Critical Exponents In Multidimensional Spaces

In this thesis,by the Mountain Pass Lemma and the Brezis-Lieb Lemma,we consider the following Kirchhoff type problem involving a critical nonlinearity: where a,b,m,λ>0 and Ω(?)RN(N≥3)is a smooth bounded domain with smooth boundary (?)Ω.We make the following assumptions on f:(f1) f is continuous in Ω×R,f(x,t)≥0 as t≥0 and f(x,t)=0 as t≤0 for all x∈Ω.(f4)There exists a nonempty open set ω ∈ Ω such that uniformly for x ∈ω.(f5)There exist constants η>0,μ>0 such that f(x,t)≥ηt for x ∈ω and for all t ∈[μ,+∞),where ω is some nonempty open subset of Ω.(f6)There exists a constant η>0 such that f(x,t)≥ηfor x∈ω and for all t ∈A, where A (?)(0,+∞)is some nonempty open interval and ω is some nonempty open subset of Ω.And then,as λ=1,we have the following results:Theorem 1 Suppose N=3,a>0,b≥0 and 0<m<2.If(f1),(f2),(f3) and (f4) hold ,problem (1) has a positive ground state solution.Corollary 1 Let a,b>0 and m=1.Assume that assumptions(f1),(f2),(f4) are sarisfied and Then problem (1) has a positive ground state solution.Theorem 2 Suppose=4,n>0,a≥0,b≥o and 0<m<1.If(f1)-(f3) and (f5) hold,problem (1) has a positive ground state solution.Theorem 3 Supppose N≥5,a>0,b≥0 and 0<m<N-2/2. If(f1)-(f3) and (f6) hold,problem (1) has a positive ground state solution.In addition,we have the following result as λ small enough:Theorem 4 Suppose=3,a>0,b>0 and m=1.If(f1),(f2)and (f4) hold,there erists a constant λ*>0 such that problem (1) has at least a positive solution for λ∈(0,λ*).