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

Abstract

We investigate entire radial solutions of the semilinear biharmonic equation Δ 2 u = λ exp ( u ) in R n , n ⩾ 5 , λ > 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 | x | . Next, we prove the existence of infinitely many entire regular radial solutions. They all diverge to −∞ as | x | → ∞ and we specify their asymptotic behaviour. As in the case with power-type nonlinearities [F. Gazzola, H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann. 334 (2006) 905–936], the entire singular solution x ↦ − 4 log | x | 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 .