The Application Of Variational Method In Several Kinds Of Kirchhoff Type Elliptic Equations

In the thesis, we use variational methods and some analysis technologies to investigate the existence or multiplicity of positive solutions for three kinds of Kirchhoff-type elliptic equations in the whole space.In Chapter 1, we introduce physical background and research status on Kirchhoff-type elliptic equations, give some necessary theorems and lemmas, and make provision for the structure of the article.In Chapter 2, we study the following Kirchhoff-type elliptic equation with zero mass and critical term nontrivial. Under appropriate conditions on a, b,u, we obtain that existence, multiplicity and nonexistence of positive solutions.In Chapter 3, we study the following Kirchhoff-type elliptic equation with general subcritical term where a>0,b>0, λ≥0 and g satisfies the general subcritical growth condition. When λ is small, we obtain equation has a positive solution, which expands the result of Li et al. [Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations 253 (2012), no.7, 2285-2294.]. When λ is big, we obtain equation has no nontrivial solution.In Chapter 4, we study the following Kirchhoff-type elliptic equation with subcritical or critical term where a> 0,b> 0,4< q≤ 6, g satisfies the subcritical growth condition and V, K, g are asymptotically periodic functions on x. Firstly, we obtain equation, where 4< q< 6, has a positive ground state solution. Secondly, by letting q→6,we obtain equation,where q=6,has a positive ground state solution.