### 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 $\mu=\mu^*$.

#### Article information

Source
Adv. Differential Equations Volume 2, Number 3 (1997), 361-382.

Dates
First available in Project Euclid: 23 April 2013