几类Kirchhoff型椭圆型方程多峰解的存在性研究

Considered in this project are several classes of Kirchhoff type non-local elliptic problems arising from nonlinear analysis and partial differential equations. The existence of multi-peak solutions for three classes of Kirchhoff problems will be obtained by combining the variational principle with gluing method and this kind of result can reflect the non-local property of Kirchhoff problems. We first construct, for a nonhomogenous Kirchhoff equation with the critical Sobolev exponent on a bounded domain with small holes, a multi-peak solution by Lyapunov-Schmidt reduction method. Secondly, we search the critical points for the energy functional associated with a Kirchhoff equation with a non-radial potential, whose locations are near a certain class of approximate solutions determined by a mini-max scheme. Last, the existence of nontrivial normalized solutions for a Kirchhoff system will be studied, and then they will be used to construct multi-peak solutions for a Kirchhoff equations with multi-well potentials. The new ideas and techniques used in this project make it possible for us to deal with the Kirchhoff terms and increase our knowledge as to the Kirchhoff type problems.

本项目考察非线性泛函分析与偏微分方程研究中出现的几类Kirchhoff型非局部椭圆型问题。申请人将结合粘接思想与变分方法来研究三类Kirchhoff问题多峰解的存在性，此类结果能反映问题的非局部性质。首先，针对一类有小洞的有界区域上带Sobolev临界和非齐次非线性项的Kirchhoff方程，我们应用Lyapunov-Schmidt约化方法构造该问题的多峰解。其次，针对一类全空间上带非对称位势的Kirchhoff方程，我们应用变分方法确定一族多峰的近似解并在其附近应用形变引理寻找问题的临界点。最后，针对一类带多位势阱的Kirchhoff约束变分问题，我们先研究一类Kirchhoff方程组对应的约束变分问题的极小元，并用其来构造原问题的多峰解。本项目在研究思路与技术上的创新能克服非局部Kirchhoff项给数学处理带来的困难，增进我们对Kirchhoff型非局部问题的认识。

本项目考察非线性泛函分析与偏微分方程研究中出现的几类Kirchhoff型非局部椭圆型问题。.申请人将结合粘接思想与变分方法来研究三类Kirchhoff问题多峰解的存在性，此类结果能反映问题的非局部性质。首先，针对一类有小洞的有界区域上带Sobolev临界和非齐次非线性项的Kirchhoff方程，我们应用Lyapunov-Schmidt约化方法构造该问题的多峰解。其次，针对一类全空间上带非对称位势的Kirchhoff方程，我们应用变分方法确定一族多峰的近似解并在其附近应用形变引理寻找问题的临界点。最后，针对一类带多位势阱的Kirchhoff约束变分问题，我们先研究一类Kirchhoff方程组对应的约束变分问题的极小元，并用其来构造原问题的多峰解。本项目在研究思路与技术上的创新能克服非局部Kirchhoff项给数学处理带来的困难，增进我们对Kirchhoff型非局部问题的认识。