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

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

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