On the elliptic equation \(\Delta u=\phi (x)e^ u\) in the plane (Q1062203)

The author studies the equation \(\Delta u=\phi (x)e^ u\), \(x\in {\mathbb{R}}^ 2\), with a locally Hölder continuous function \(\phi\). By an entire solution we mean a function \(u\in C^ 2({\mathbb{R}}^ 2)\) which satisfies the above equation at every point \(x\in {\mathbb{R}}^ 2\). \textit{W. M. Ni} [Invent. Math. 66, 343-352 (1982; Zbl 0487.35042)] has founded conditions for entire solutions with various orders of growth at infinity. In this paper the author looks for conditions garanteeing the existence of entire solutions which are eventually positive and have logarithmic growth at infinity. His method relies on the results of Kawano, Kusano and Naito (Proc. Amer. Math. Soc. to appear) who studied the equation \(\Delta u=\phi (x)u^{\gamma}\), \(x\in {\mathbb{R}}^ 2\), \(\gamma\) a positive constant.