Four positive solutions for the semilinear elliptic equation: $-\Delta u+u=a(x)u^p+f(x)$ in ${\mathbb R}^N$

Abstract.

We consider the existence of positive solutions of the following semilinear elliptic problem in ${\mathbb R}^N$:

$\aligned -\Delta u + u &= a(x)u^p + f(x)\qquad in {\mathbb R}^N, \cr u &>0\qquad \qquad \qquad \quad in {\mathbb R}^N, \cr u &\in H^1({\mathbb R}^N), \cr \endaligned \eqno(*)$

where $\displaystyle 1 < p < {{N+2}\over{N-2}} (N\geq 3)$, $1< p < \infty (N=1, 2)$, $a(x)\in C(\mathbb R}^N)$, $f(x)\in H^{-1}({\mathbb R}^N)$ and $f(x)\geq 0$. Under the conditions:

1°$a(x)\in (0,1]$ for all $x\in{\mathbb R}^N$,

2°$a(x)\rightarrow 1$ as $|x|\rightarrow \infty$,

3° there exist $\delta>0$ and $C>0$ such that

$a(x)-1 \geq -C e^{-(2+\delta)\abs x} \qquad for all x\in{\mathbb R}^N,$

4°$a(x)\not\equiv 1$,

we show that (*) has at least four positive solutions for sufficiently small $\|f\|_{H^{-1}({\mathbb R}^N)}$ but $f\not\equiv 0$.