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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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 \(f\not\equiv 0\).