带有临界增长的Kirchhoff型问题的基态解

Ground State Solutions for Some Kirchhoff Type Problems with Critical Growth in R^3

本文主要考虑如下Kirchhoff问题{-（a＋b∫R3｜^？u｜^2dx）？u＋u=f（x,u）＋Q（x）｜u｜^4u,u∈H^1（R^3）,其中a,b是正的常数.我们证明了基态解,即上述问题的极小能量解的存在性.同时,如果假定Q≡1,且h（x）满足一定的条件,可以证明下述问题{-（a＋b∫R3｜？u｜^2dx）？u＋u=｜u｜^4u＋h（x）u,u∈H^1（R^3）的基态解的存在性.

We are concerned with a class of Kirchhoff problem{-（a＋b∫R3｜？u｜^2dx）？u＋u=f（x,u）＋Q（x）｜u｜^4u,u∈H^1（R^3）,where a, b 0 are constants. The existence of ground state solutions, i.e., nontrivial solutions with least possible energy of this Kirchhoff problem is obtained. Moreover,when Q 三 1,under suitable conditions on h（x）,we prove the existence of ground state solutions for the the following Kirchhoff problem{-（a＋b∫R3｜？u｜^2dx）？u＋u=｜u｜^4u＋h（x）u,u∈H^1（R^3）