质量临界非齐次薛定谔方程在门槛值处的极限行为

Limit Behavior of Mass Critical Inhomogeneous Schrodinger Equation at the Threshold

该文研究如下与时间无关的具有吸引相互作用的临界非齐次薛定谔方程-△u+|x|^(2)u-am(x)|u|4/Nu=μu,in RN,N≥1,其中a> 0且0 0及适合的0≤g(x)<1,令m(x)=1-λg(x),证明该方程在阈值a=a*处基态解的存在性,并给出λ→0+时基态解的极限行为.这些结论推广了Deng,Guo和Lu^([10,11])的结果.特别地,该文使用了一种直接而更简单的方法得到能量的下界.

This paper is concerned with the following time-independent critical inhomogeneous Schrodinger equation with attractive interactions:-△u+|x|^(2)u-am(x)|u|4/Nu=μu,in RN,N≥1.where a> 0,0*for m(x)= 1-λg(x) with λ> 0 and suitable 0 ≤g(x) <1,and then investigate the limit behavior of those threshold ground states as λ→0+.These conclusions extend the results of Deng-Guo-Lu^([10,11]).In particular,compared to the arguments of ^([10,11]),we use a direct and simpler method to obtain the lower bound of energy.