A note on the implicit function theorem for quasi-linear eigenvalue problems

We consider the quasi-linear eigenvalue problem −Δpu=λg(u)−Δpu=λg(u) subject to Dirichlet boundary conditions on a bounded open set ΩΩ, where gg is a locally Lipschitz continuous function. Imposing no further conditions on ΩΩ or gg, we show that for λλ near zero the problem has a bounded solution which is unique in the class of all small solutions. Moreover, this curve of solutions parameterized by λλ depends continuously on the parameter.