We consider a nonlinear elliptic boundary value problem on a planar domain. The exponential type nonlinearity in the boundary condition is one that frequently appears in the modeling of electrochemical systems. For the case of a disk we construct a family of exact solutions that exhibit limiting logarithmic singularities at certain points on the boundary. Based on these solutions we develop two criteria that we believe predict the possible locations of the boundary singularities on quite general domains.
Bryan, Kurt M. and Vogelius, Michael, "Singular solutions to a nonlinear elliptic boundary value problem originating from corrosion modeling" (2000). Mathematical Sciences Technical Reports (MSTR). 97.