Singular solutions of semilinear elliptic equations with exponential nonlinearities in 2-dimensions

By introducing a new classification of the growth rate of exponential functions, singular solutions for − Δ u = f ( u ) in 2-dimensions with exponential nonlinearities are constructed. The strategy is to introduce a model nonlinearity “close” to f , which admits an explicit singular solution. Then, using a transformation as in [8] , one obtains an approximate singular solution, and then one concludes by a suitable fixed point argument. Our method covers a wide class of nonlinearities in a unified way, e.g., f ( u ) = u r e u q ( q > 1 , r ∈ R ) , f ( u ) = e u q + u r ( q > 1 , q / 2 > r > 0 or 1 < q < 4 , r = q − 1 ), f ( u ) = e e u . As a special case, our result contains a pioneering contribution by Ibrahim–Kikuchi–Nakanishi–Wei [15] for u ( e u 2 − 1 ) .