The asymptotic behaviour of the $ p(x) $-Laplacian Steklov eigenvalue problem

In this paper we study the asymptotic behaviour of the first eigenvalues $ \lambda^{1}_{p_{n}(\cdot)} $ and the corresponding eigenfunctions $ u_{n} $ of (1) as $ p_{n}(x)\rightarrow \infty $. Under adequate hypotheses on the sequence $ p_{n} $, we prove that $ \lambda^{1}_{p_{n}(\cdot)} $ converges to 1 and the positive first eigenfunctions $ u_{n} $, normalized by $ |u_{n}|_{L^{p_{n}(x)}(\partial \Omega)} = 1 $, converge, up to subsequences, to $ u_{\infty} $ uniformly in $ C^{\alpha}(\overline{\Omega}) $, for some $ 0<\alpha<1 $, where $ u_{\infty} $ is a nontrivial viscosity solution of a problem involving the $ \infty $-Laplacian subject to appropriate boundary conditions.