Multiple Solutions For Kirchhoff Type Of Problems Involving Neumann Boundary Condition

In this thesis,we investigate multiplicity of solutions for two kinds of Kirch-hoff type equation involving Neumann boundary condition,via the Symmetric moun-tain pass lemma,the Nehari manifold and the concentration compactness principle.Firstly,we study the following Kirchhoff type equation:where Ω is smooth bounded domain in R3,a,b>0 are two real parameters,υ denotes the derivative along the outer normal,and f:Ω × R1 →R1 is a Caratheodory function with subcritial growth condition.Under certain assumptions on the function f(·)and c(·),we obtain the existence of infinitely many solutions of Kirchhoff equation by using the the Symmetric moun-tain pass lemma.Next,we study the following Kirchhoff type equation involving the critical growth term:where Ω is smooth bounded domain in R3,1<q<2,ε>0 is enough small.υdenotes the derivative along the outer normal.The weight functions fλ,defined by fλ=λf + f—,λ>0 is positive real number,with f± = ± max{±f,0}(?)0 andWe prove that the problem has at least two positive solutions,and one of solution is a positive ground sate solution by the Nehari manifold,the concentration compace-ness principle and some analytical techniques.