Isolated singularities of solutions of linear and semilinear elliptic equations with singular drifts

Abstract

We study isolated singularities of solutions of linear and semilinear elliptic equations in divergence form with singular drifts. First, extending a classical result for isolated singularities of harmonic functions, we establish a removable isolated singularity theorem for linear equations with drifts b in $L^{q_0}\left(B_R\text{; }{R}^n\right)$ for some $q_0\ge n$, where $n\ge 2$ is the dimension. Then this theorem is applied to prove removability theorems for isolated singularities of solutions of some semilinear equations with drifts in $L^{q_0}\left(B_R\text{; }{R}^n\right)$. One novelty of our results is that the critical case $q_0=n$ is allowed for removable singularity theorems for both linear and semilinear equations. Moreover, our methods of proofs rely only on interior $W^{1,q}$-estimates for solutions on annuli and $L^q$-estimates for their traces on spheres but not pointwise estimates like the maximum principle, which can be thus applied to linear and nonlinear systems.