Nonlinear boundary value problems relative to harmonic functions

Abstract

We study the problem of finding a function u verifying −Δu=0 in Ω under the boundary condition ∂u∂n+g(u)=μ on ∂Ω where Ω⊂RN is a smooth domain, n is the normal unit outward vector to Ω, μ is a measure on ∂Ω and g a continuous nondecreasing function. We give sufficient condition on g for this problem to be solvable for any measure. When g(r)=|r|p−1r, p>1, we give conditions in order an isolated singularity on ∂Ω to be removable. We also give capacitary conditions on a measure μ in order the problem with g(r)=|r|p−1r to be solvable for some μ. We also study the isolated singularities of functions satisfying −Δu=0inΩ and ∂u∂n+g(u)=0 on ∂Ω∖{0}.