Existence And Multiplicity Of Solutions For The Kirchhoff Type Problems With Critical Nonlinearity

In this paper,we mainly study two kinds of Kirchhoff type problems with critical non-linearity by means of variational methods,we obtain the existence of nontrivial solutions and multiple solutions respectively under suitable conditions.The thesis consists of four chapter-s.In the first chapter,we mainly introduce the background and significance of modern variational methods and Kirchhoff type problems and we also introduce the problem and main work of this dissertation.In the second chapter,we give the related preparatory knowledge.In Chapter 3,we consider the existence of solutions for the following Kirchhoff type problem involving p-biharmonic operators where ? is a bounded domain in RN with C2 boundary,N?2,? is the Laplace operator and(?)is the outer normal derivative,? is a positive parameter,f:?×R?R is a Caratheodory function,p**is the critical exponent,i.e.where p ?(2,+?).Based on the symmetric mountain pass theorem due to Kajikiya and the Concentration Compactness Principle due to Lions,we obtain the following conclusion:there exists ?*>0 such that for all ??(0,?*),problem(1)has a nontrivial solution.Morover,if n?+?,un?0.In Chapter 4,we consider the existence and multiplicity of solutions for the following fourth-order elliptic equations of Kirchhoff type where a,b>0,2**=2N/N-4 is the critical exponent according to the Sobolev embedding.k(x)?Lr(RN),r=2**/2**-q,? and ? are real parameters.N?5.By using Lion's second Concentration compactness principle and Concentration com-pactness principle at infinity to prove that(PS)c condition holds locally and by minimax methods and Krasnoselski genus theory,we establish the multiplicity of solutions.