Invariant criteria for existence of bounded positive solutions

We consider semilinear elliptic equations Δu ± p(x) f(u) = 0, or more generally Δu + φ(x, u) = 0, posed in R^N (N ≥ 3). We prove that the existence of entire bounded positive solutions is closely related to the existence of bounded solution for Δu + φ(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 applied for the existence of large solutions, for the exterior domain problem and for the system situations.