Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity

[摘要] We investigate entire radial solutions of the semilinear biharmonic equation Delta(2)u = lambda exp(u) in R-n >= 5, lambda > 0 being a parameter. We show that singular radial solutions of the corresponding Dirichlet problem in the unit ball cannot be extended as solutions of the equation to the whole of R-n. In particular, they cannot be expanded as power series in the natural variable s = log vertical bar x vertical bar. Next, we prove the existence of infinitely many entire regular radial solutions. They all diverge to -infinity as vertical bar x vertical bar -> infinity and we specify their asymptotic behaviour. As in the case with power-type nonlinearities [E Gazzola, H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann. 334 (2006) 905-936], the entire singular solution x -> -4 log vertical bar x vertical bar plays the role of a separatrix in the bifurcation picture. Finally, a technique for the computer assisted study of a broad class of equations is developed. It is applied to obtain a computer assisted proof of the underlying dynamical behaviour for the bifurcation diagram of a corresponding autonomous system of ODEs, in the case n = 5. (c) 2006 Elsevier Inc. All rights reserved.

[发布日期] 2006-11-15

[关键词] radial solutions;biharmonic equation;computer assisted proof;supercritical growth;dynamical system;separatrix