Existence of multiple positive solutions for a semilinear elliptic equation

Abstract

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 $\mu=\mu^*$.