Study On The Existence And Multiplicity Of Solutions Of Two Kinds Of Kirchhoff Type Equations

We investigate the existence and multiplicity of solutions for two classes of Kirchhoff type equations in R N using the variational methods.The thesis consists of four chapters. Chapter 1 is the introduction.In chapter 2, we firstly study the following Kirchhoff equations where a > 0, b > 0 and N ≥ 3. Under suitable conditions on K and f, we obtain four existence results and two nonexistence results by using monotonicity trick and Pohozaev identity. Secondly, when K(x)f(u) = f(u) + h(x)|u|q-2u, we are concerned with a class of the following Kirchhoff type equations with zero mass where a > 0 is a constant, 1 < q < 2, λ > 0 is a parameter. If N ≥ 3, we give the result of multiplicity of two positive solutions via the variational method and Pohozaev identity; if N ≥ 5, the result of multiplicity of three positive solutions is established.In chapter 3, we consider the multiplicity of positive solutions for a class of superlinear Kirchhoff type problems where N ≥ 3, 1 < q < 2, a > 0, b > 0, λ ≥ 0 is a parameter, and m, f, h are positive continuous functions. Using the variational method and iterative techniques, we show that if the nonlinear is subcritical and superlinear at zero and infinity, the Kirchhoff type problems admits at least two positive solutions when the parameter is sufficiently small. Our result generalizes some recent results about Kirchhoff type problems.In chapter 4, we study the asymptotically linea Kirchhoff equations where N ≥ 3, a > 0, λ > 0 is a parameter, and the nonlinear f is a continuous function. The multiplicity of positive solutions for Kirchhoff type equation proved by Mountain pass theorem and Ekeland’s variational principle. We will show that if the nonlinear is asymptotically linear at infinity and λ > 0 is sufficiently small, the Kirchhoff type equations has at least two positive solutions.For the perturbed problem, we give the result of multiplicity of three positive solutions.