Solvability of the Neumann problem in a ball for \(-\Delta u + u^{-\nu} = h(| x |), \nu > 1\)
The authors obtain existence and uniqueness results for radially symmetric solutions of the elliptic problem \(- d\Delta u+ u^{- \nu}= h\) on \(B\) with zero Neumann boundary condition \((\nu> 1, d> 0)\). \(B\) is the unit ball in \(\mathbb R^N\) with \(N\geq 2\). Assuming that the (radial) function \(h\) satisfies \(\int_B h\, dx> 0\) they obtain the existence of a solution. With the additional assumption that \(d\) is sufficiently large the solution is shown to be unique. Existence of a positive solution follows for the integral being strictly positive.
scientific article; zbMATH DE number 838712