中文摘要：

该文讨论如下问题的正解的存在性与非存在性（P）{-△u＋V（x）u＋λФ（x）｜u｜^q-2u=f（u） in R^3 -△Ф=｜u｜^q1lim ｜x｜→＋∞ Ф（x）=01 lim｜x｜→＋∞ u（x）=0,这里λ〉0是参数,5/3≤q〈5,f（s）在无穷远处关于s是渐进线性的.在较简单的假设下,我们证明了：在λ较小时,问题（P）至少有一个正解,在λ较大时,问题（P）不存在非平凡解.

英文摘要：

In this paper,we investigate the existence and non-existence of positive solutions of the following problem （P）{-△u＋V（x）u＋λФ（x）｜u｜^q-2u=f（u） in R^3 -△Ф=｜u｜^q1lim ｜x｜→＋∞ Ф（x）=01 lim｜x｜→＋∞ u（x）=0,whereλ0 is a parameter,5/3≤q 5,f（s） is asymptotically linear with respect to s at infinity. Under some simple assumptions on V and f,we prove that the problem（P） has at least one positive solution forλsmall and has no nontrivial solution forλlarge.