Nonlinear boundary value problems relative to harmonic functions

We study the problem of finding a function u verifying -Delta u = 0 in Omega under the boundary condition partial derivative u/partial derivative n + g(u) = mu on partial derivative Omega where Omega subset of R-N is a smooth domain, n is the normal unit outward vector to Omega, mu is a measure on partial derivative Omega and g a continuous nondecreasing function. We give sufficient condition on g for this problem to be solvable for any measure. When g(r) = vertical bar r vertical bar(p-1)r, p > 1, we give conditions in order an isolated singularity on partial derivative Omega to be removable. We also give capacitary conditions on a measure mu in order the problem with g(r) = vertical bar r vertical bar(p-1)r to be solvable for some mu. We also study the isolated singularities of functions satisfying -Delta u = 0 in Omega and partial derivative u/partial derivative + g(u) = 0 on partial derivative Omega\{0}. (C) 2020 Elsevier Ltd. All rights reserved.