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

We consider the quasi-linear eigenvalue problem $-{\mathrm{\Delta }}_{p}u=\lambda g(u)$ subject to Dirichlet boundary conditions on a bounded open set $\mathrm{\Omega }$, where $g$ is a locally Lipschitz continuous functions. Imposing no further conditions on $\mathrm{\Omega }$ or $g$ we show that for small $\lambda$ the problem has a bounded solution which is unique in the class of all small solutions. Moreover, this curve of solutions depends continuously on $\lambda$.