, Volume 11, Issue 1, pp 63-95

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

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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:

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

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

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

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

$$a(x)\not\equiv 1$$,

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