Remarks on entire solutions for two fourth order elliptic problems

We are interested in entire solutions for the semilinear biharmonic equation ∆u = f(u) in R , where f(u) = e or −u−p (p > 0). For the exponential case, we prove that for the polyharmonic problem ∆u = e with positive integer m, any classical entire solution verifies ∆2m−1u < 0, this completes the results in [6, 14]; we obtain also a refined asymptotic expansion of radial separatrix solution to ∆u = e in R, which answers a question in [2]. For the negative power case, we show the nonexistence of the classical entire solution for any 0 < p ≤ 1. Mathematics Subject Classification (2000): 35J91, 35B08, 35B53, 35B40.