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 \(f\not\equiv 0\).
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Received December 11, 1998 / Accepted July 16, 1999 / Published online April 6, 2000
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Adachi, S., Tanaka, K. Four positive solutions for the semilinear elliptic equation: \(-\Delta u+u=a(x)u^p+f(x)\) in \({\mathbb R}^N\) . Calc Var 11, 63–95 (2000). https://doi.org/10.1007/s005260050003
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DOI: https://doi.org/10.1007/s005260050003