Remarks on Entire Solutions for Two Fourth-Order Elliptic Problems

We are interested in entire solutions for the semilinear biharmonic equation δ²u = f(u) in RN, where f(u) = e^u or .u^-p (p > 0). For the exponential case, we prove that for the polyharmonic problem Ä2mu = eu with positive integer m, any classical entire solution verifies δ^2m-1u < 0; this completes the results of Dupaigne et al. (Arch. Ration. Mech. Analysis 208 (2013), 725-752) and Wei and Xu (Math. Annalen 313 (1999), 207-228). We also obtain a refined asymptotic expansion of the radial separatrix solution to ²u = eu in R³, which answers a question posed by Berchio et al. (J. Diff. Eqns 252 (2012), 2569-2616). For the negative power case, we show the non-existence of the classical entire solution for any 0 < p≤1.