一类p-Kirchhoff方程基态解的存在性与唯一性

Existence and Uniqueness of Ground State Solutions for a Class of p-Kirchhoff Equations

对于下面p-Kirchhoff型泛函Ep(u)=a/p∫Rn|■u|pdx+b/2p(∫Rn|■u|pdx)2-1/s∫Rn||u|sdx,我们证明了约束在流形Sc:={u∈W1,p(Rn):∫Rn}|u|pdx=cp}上全局极小点或山路型临界点的存在性与唯一性,且这些临界点是某个Gagliardo-Nirenberg不等式的最优化子,特别当p∈(1,2]时,它们在不计平移意义下是唯一的.我们扩展了已有文献中p=2的情形的相关结果.

For the following p-Kirchhoff type functional Ep(u)=a/p∫Rn|■u|pdx+b/2p(∫Rn|■u|pdx)2-1/s∫Rn||u|sdx,we prove the existence and uniqueness of global minimum or mountain pass type critical points on the Lp-normalized manifold Sc:={u∈W1,p(Rn):∫Rn|u|pdx=cp}.We show that these critical points indeed are optimizers of a certain Gagliardo-Nirenberg inequality.Especially,when p∈(1,2],they are unique up to translations.We extend some known results for the case of p=2 in previous papers.