Asymptotic behavior of solutions to elliptic equations in 2D exterior domains

The asymptotic behavior of solutions to second order elliptic equations in two-dimensional exterior domains is studied. In particular, under the assumption that the solution belongs to the Lorentz space $$L^{p,q}$$ or the weak Lebesgue space $$L^{p,\infty }$$ with certain conditions on the coefficients, we give a natural and the optimal sharp pointwise estimate of the solution at spatial infinity. The proof is based on the level set approach of solutions introduced by Korobkov et al. (Arch Rat Mech Ann 233:385–407, 2019), in which the decay property of the solution to the vorticity equation of the two-dimensional Navier–Stokes equations was studied.