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We consider semilinear elliptic equations ∆u ± ρ(x)f (u) = 0, or more generally ∆u + ϕ(x, u) = 0, posed in R N (N ≥ 3). We prove that the existence of bounded positive entire solutions is closely related to the existence of bounded solution for ∆u + ρ(x) = 0 in R N . Many sufficient conditions which are invariant under the isometry group of R N are established. Our proofs use the standard barrier method, but our results extend many earlier works in this direction. Our ideas can also be applieddoi:10.3934/dcds.2005.12.413 fatcat:5viamud5qzhkri5bmnodw7pcou