Department of Mathematics, Huazhong Normal University, Wuhan, 430070, China

Department of Mathematics, University of Rochester, Rochester, NY 14627, United States

Department of Mathematics, University of Iowa, Iowa City, IA 52242, United States

[Yinbing Y.] Department of Mathematics, Huazhong Normal University, Wuhan, 430070, China, Department of Mathematics, University of Rochester, Rochester, NY 14627, United States

[Li Y.] Department of Mathematics, University of Rochester, Rochester, NY 14627, United States, Department of Mathematics, University of Iowa, Iowa City, IA 52242, United States

In this paper;we consider the semilinear elliptic problem $$ -\triangle u+ u=|u|^{p-2}u+ \mu f(x);\quad u \in H^1(\Bbb R^N);\quad N>2. \tag"$(*)_\mu$" $$ For $p> 2$;we show that there exists a positive constant $\mu ^*>0$ such that $(*)_\mu$ possesses a minimal positive solution if $\mu \in (0;\mu ^*)$ and no positive solutions if $\mu > \mu^*$. Furthermore;if $p< \frac{2N}{N-2}$;then $(*)_\mu$ possesses at least two positive solutions for $\mu \in (0;\mu^{*})$;a unique positive solution if $\mu =\mu^*$ and there exists a constant $\mu _{*} >0 $ such that when $ \mu\in (0;\mu_{*})$;problem $(*)_\mu$ possesses at least three solutions. We also obtain some bifurcation results of the solutions at $\mu =0$ and $\mu=\mu^*$. Published: 1997 First available in Project Euclid: 23 April 2013 zbMATH: 1023.35503 MathSciNet: MR1441848 Digital Object Identifier: 10.57262/ade/1366742248 Subjects: Primary: 35J60 Secondary: 35B05;35B32

数学与统计学学院