Existence Of Solutions For Two Types Of Kirchhoff Differential Equations

In this thesis, we mainly consider the existence of solutions for two types of Kirchhoff differential equations. Firstly, by using bifurcation arguments and topological degree theory, we discuss the following non-homogenous Kirchhoff type equation where Ω(?)R3 is bounded, in the condition of that f(x,u) is asymptotical linear at zero and asymptotically 3-linear at infinity, or f is subcritical linear at zero, we give the sufficient condition of the existence of positive solutions. The result we get extends the main result of [Liang, et al, Ann. I. H. Poincare-AN(2014)] in which a(x), b(x) given are constant functions, and the main result of [Figueiredo, et al, J. Math. Anal. Appl. (2014)], where f(x,u)=u~q.Next, we study a class of forth order Kirchhoff type equation which is defined in RN, depending on a function m and a nonnegative parameter λ by using variational methods, iterative technique and Pohozaev type identity, we investigate the existence and nonexistence of nontrivial solutions of the above equation. The result we get extends the main result of [Wang, et al, J. Math. Anal. Appl. (2014)].