Research On Positive And Sign Solutions Of Kirchhoff Elliptic Equations

In this thesis,by using constraint variational methods and some analysis techniques,we study the existence of ground state solutions or ground state sign-changing solutions,the multiplicity of positive solutions,the convergence property of ground state sign-changing solutions for some kinds of Kirchhoff type problems.Firstly,we consider the following Kirchhoff type problem with a critical exponent(?)where ? is a smooth bounded domain in R3,b is a positive constant,?>?1 is a positive parameter,?1 is the principal eigenvalue of the operator-? under Dirichlet boundary conditions,and the nonlinear growth of |u|4u reaches the Sobolev critical exponent since the critical exponent 2*= 6 in three spatial dimensions.With the help of the Nehari manifold and the Brezis-Lieb lemma,we obtain the multiplicity of positive solutions and the existence of positive ground state solutions for ? in a small right neighborhood of ?1 and all b>0.Secondly,we investigate the following Kirchhoff type problem on R3 involving a critical nonlinearity(?)where b is a positive constant,?>?1 is a positive parameter,?1 is the principal eigenvalue of-?u=?f(x)u,u?D1,2(R3),D1,2(R3)= {u ? L2*(R3):?u ?L2(R3)},the weight function f is nonnegative and nonzero.We obtain the multi-plicity of positive solutions and the existence of ground state solutions.Our tools are the Nehari manifold and the concentration compactness principle.Thirdly,by using of the Weierstrass theorem and the Nehari manifold,we study the multiplicity of positive solutions for the following Kirchhoff type prob-lem with indefinite nonlinearities where a,b>0,?>?1,?1 is the principal eigenvalue of-?u + u = ?f(x)u,u? H1(RN),f is nonnegative and nonzero,g is sign-changing in RN.Next,we are concerned with the following Kirchhoff type problem on RN with indefinite nonlinearities where b,b are positive constants,1<r<2,4<s<2*= 2N/N-2,? is a positive parameter,the weight function f is nonnegative and nonzero,g changes sign.By the Nehari manifold,we obtain the multiplicity of positive solutions and the existence of ground state solutions.Finally,we investigate the existence and nonexistence of ground state sign-changing solutions for the following Kirchhoff-type problem where ? is a smooth bounded domain in RN,= 1,2,3,a,b>0,?<a?1.With the help of the sign-changing Nehari manifold and some analysis techniques,we not only prove the existence and convergence property of ground state sign-changing solutions for all 0<b<?(?>0),but also prove the nonexistence result of sign-changing solutions for all b>?.