On anisotropic singularities for semi-linear elliptic equations in R2

In this note, we study weak solutions of(1)−Δu+g(u)=8π∂δ0∂x2inR2,−Δ+gu(x)→0as|x|→+∞, where g(s)=2es−1es+1 is an odd, increasing and bounded function and ∂δ0∂x2 is defined in the sense of distribution〈∂δ0∂x2,ξ〉=∂ξ(0)∂x2forξ∈Cc1(R2). We obtain that problem (1) admits a unique, x2-odd weak solution, which has anisotropic singularity 4x2|x|2 near the origin.