非局部问题在不同无界域下的古典解

Classical Solutions for Nonlocal Problems on Different Unbounded Domains

本文在不同的无界域上考虑了一类含线性到临界增长的非局部问题古典解的存在性,立足于构造函数的思想,给出问题无穷多古典正解的具体形式.首先,基于最佳Sobolev嵌入常数所对应的达到函数,获得了临界增长情形该问题在全空间上的无穷多古典解;其次,利用分离变量法在无坐标平面的半空间上获得相同的结论并且在无坐标平面的全空间也成立;最后,证明了在无坐标平面的全空间上满足线性到临界之间增长时也有无穷多古典解.

This paper address the existence of classical solutions for a kind of nonlocal problems with linear to critical exponents on different unbounded domains,and the expressions of infinitely many classical positive solutions are given which basing on the idea of constructors.First of all,infinitely many classical solutions are obtained by using the Sobolev's achieving function of the best constant on while space with critical growth.Next,we gain the same conclusions on half-space and all space with the help of separation variable methods without the Cartesian coordinate planes.Finally,we prove that the results are suitable for all exponents between linear and critical on all space without the Cartesian coordinate planes.