Entire Solutions for a Semilinear Fourth Order Elliptic Problem with Exponential Nonlinearity∗

We investigate entire radial solutions of the semilinear biharmonic equation ∆u = λ exp(u) in Rn, 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 Rn. 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 [GG], the entire singular solution x 7→ −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. Mathematics Subject Classification: 35J60; 35J30, 35J65, 35 J40.