【作者】 冯仁婷；

【导师】 唐春雷；

【作者基本信息】 西南大学 ， 基础数学， 2022， 硕士

【摘要】 本文主要研究Kirchhoff型方程解的存在性.首先,考虑如下Kirchhoff型问题:(?)其中a,b>0,V是位势,f为非线性项.我们对f提出了如下假设:(f1)f(t)∈C(R3,R),且(?)f(t)/t=0.(f2)(?)f(t)/t3=α,其中 α ∈(0,+∞).(f3)函数f(t)/|t|3在(-∞,0)和(0,+∞)上是严格单调递增的.我们先研究问题(0.0.1)的变号解的存在性和渐近性.运用Nehari流形和形变引理,得到了问题(0.0.1)的仅变号一次的基态变号解,且证明到其能量严格大于基态解能量的两倍.此外,当b → 0时,我们证到问题(0.0.1)的基态变号解渐近到如下Sch(?)dinger方程的基态变号解:-a△u+V(x)u=f(u).(0.0.2)接着,考虑如下带凹凸非线性项的Kirchhoff型问题:其中a,b>0,1<q<2,V是位势,g(x)∈Lq*,q*=2/2-q,且g(x)>0.基于第二章的研究内容,我们考虑当f在无穷远处满足渐近3-线性增长时,问题(0.0.3)正解的存在性.仍在假设条件(f1)-(f3)成立的基础上,我们证明到问题(0.0.3)对应的能量泛函具备山路结构,再进一步证到其满足Palais-Smale条件,最后通过局部极小原理和山路定理得到问题(0.0.3)的两个正解.更多还原

【Abstract】 This paper mainly studies the existence of solutions of Kirchhoff-type equations.Firstly,we consider the following Kirchhoff-type problem:(?)where a,b are positive constants,V is a potential,f is the nonlinearity.We make the following assumptions about f:(f1)f(t)∈ and f(t)=o(|t|)as |t|→ 0.(f2)f(t)/t3→a∈(0,+∞)as |t|→∞.(f3)f(t)/|t|3 is increasing for |t|>0.Firstly,we study the existence and asymptotic properties of the sign-changing solution of problem(0.0.1).By Nehari manifold and deformation lemma,the ground-state sign-changing solution of problem(0.0.1)with only one change of sign is obtained,and it can be proved that its energy is strictly greater than twice the energy of the ground-state solution.In addition,when b→0,we prove that the ground state sign-changing solution of problem(0.0.1)strongly converges to the ground state signchanging solution of the following Sch(?)dinger equation:-a△u+V(x)u=f(u).(0.0.2)Next,we consider the following Kirchhoff-type problem with convex and concave nonlinear terms:where a,b are positive constants.V is a potential,g(x)∈Lq*,q*=2/(2-q)and g(x)>0.Based on the research content in Chapter 2,we consider the existence of positive solutions to problem(0.0.3)when f satisfies asymptotic 3-linear growth at infinity.Still on the basis of the assumption that the condition(f1)-(f3)are established,we prove that the energy functional corresponding to the problem(0.0.3)has a mountain structure and satisfies the Palais-Smale condition.Finally,two positive solutions to the problem(0.0.3)are obtained by using the local minimum principle and the mountain pass theorem.更多还原