Existence Of Solutions For Kirchhoff-type Equations With Critical And Supercritical Growth

As an important part of modern mathematics,nonlinear partial differential equation is used to study the theoretical or practical problems in the field of natural science and engineering.With the analysis of objective facts,scholars abstract natural phenomena as mathematical models.Kirchhoff-type equation is an important kind of nonlinear PDE.In this paper,we discuss the existence of solutions for Kirchhoff-type equations by means of variational method,minimax method,truncation technique and iterative technique.This article is divided into three chapters.In the first chapter,we introduce the research background and present situation of Kirchhoff-type equations.In Chapter 2,we consider the quasilinear equations with Kirchhoff-type perturbation(1+?R3g2(u)|?u|2)[-div(g2(u)?u)+g(u)g'(u)|?u|2]+V(x)u=?|u|p-2u+f(u),x ? R3,where ?>0 is a parameter,p ?(2,6),g:R ?[0,?)is an even function,potential function V is continuous and the nonlinear term f satisfies the local superlinear growth condition,that is,(V1)V E C1(R3)is a radically symmetric and there exist V0,V?>0 such that V0?V(x)?V? for x ? R3;(V2)|?(x)·x|?1/(8|x|2)for x ? R3\{0};(f1)f ?C(R),f(t)t?0 for t ? R,and limt?o f(t)/t=0.By using variational method and Moser iterative technique of two terms,the following theorem can be obtained.Theorem 2.1.1 Assume that(V1),(V2),and(f1)hold.Then the problem(2.1.1)possesses nontrivial solutions for all sufficiently large ?.In Chapter 3,we consider the following Kirchhoff-type equations with Hartree term and critical growth nonlinearity-(a+b?R3|?u|2)?u+V(x)u+(I?*|u|p)|u|p-2u=|u|4+?f(u),x ?R3,where a>0,b?0 are constants,?>0 is a parameter,potential function V ?C(R3),? ?(0,3),I?(x):=?((3-?)/2)/2??3/2?(?/2)|x|3-?,x?R3\{0},the sign*is the convolution of the functions on R3,p ?[2,3).Suppose the potential function V and the nonlinear term f ? C1(R)satisfying(V)infx?R3 V(x)=V0>0,and for any given M>0,there exists r>0 such that(?)?(?x ?R3:|x-y|?r,V(x)?M})=0,where ?(A)is the Lebesgue measure of A;(f1)there exist q ?[2,6)and C>0 such that |f'(t)|?C(1+|t|q-2)for t ? R;(f2)limt?0 f(t)/(|t|2(p-1)t)=0;(f3)f(t)/(|t|2(p-1)t)is decreasing on(-?,0);f(t)/(|t|2(p-1)t)is increasing on(0,?).Using restricted variation methods and quantitative deformation lemmas,the following theorem can be obtained.Theorem 3.1.1 Assume that(V)and(f1)-(f3)are satisfied.Then there exist ?0>0 such that for all ?>?0,the problem(3.1.1)has least energy sign-changing solutions.