The Existence And Multiplicity Of Solutions For Two Classes Of Kirchhoff Equations With Critical Exponent

In the thesis, we consider the following Kirchhoff-type equation with concave-convex nonlinearities where a, b are positive constants, λ>0,0<q<1 and h∈L4/3-q\{0} is non-negative. When μ satisfies some conditions, we obtain the existence and multiplicity of positive solutions for the above equation by the variational method.The following Theorem is our main conclusion:Theorem 1 Suppose a>0, b> 0 and h∈L4/3-q(R4)\{0}, then1) when μ>bS2, there exists λ*> 0 such that Eq.(0.3) has two positive solutions for each λ∈G (0, λ*);2) when 0<μ≤bS2, Eq.(0.3) has at least one positive solution for all λ>0. Consider a class of Kirchhoff type problem with critical exponent in IR3:where a> 0, b> 0,λ> 0,3< q< 5. By Mountain Pass Lemma and Brezis-Lieb Lemma, we get at least one positive solution.The following Theorem is our main conclusion:Theorem 2 Suppose a, b is positive constant,3<q<5, then the equation(0.2) has at least a positive solution.