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

Abstract

By introducing a new classification of the growth rate of exponential functions, singular solutions for $-\Delta u=f\left(u\right)$ 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., $f\left(u\right)=u^r{e}^{u^q}(q>1,r\in R),f\left(u\right)=e^{u^q+u^r}$ ($q>1$, $q/2>r>0$ or $1<q<4$, $r=q-1$), $f\left(u\right)=e^{e^u}$. As a special case, our result contains a pioneering contribution by Ibrahim–Kikuchi–Nakanishi–Wei [15] for $u(e^{u^2}-1)$.