Existence of radial solutions for an elliptic problem involving exponential nonlinearities (Q939261)

The following semi-linear elliptic problem (P): \(-\Delta u + a(| x| )u= \lambda e^u\) in \(B_1\), \(u=0\) on \(\partial B_1\), where \(B_1\) is the unit ball in \(\mathbb{R}^N\), \(N\geq 2\), \(\lambda>0\) and \(a(| x| )\geq 0\) is a smooth radial function, is considered in the paper. The existence of radial solutions of (P) is proved under some suitable assumptions on the regular part of the Green function of the operator \(-u''-\frac{N-1}{r} u+a(r)u\) and \(\lambda\) small enough.