An eigenvalue problem related to the variable exponent double-phase operator

In this paper, we studied a double-phase eigenvalue problem with large variable exponents. Let $ \lambda^{1}_{(p_{n}(\cdot), \, q_{n}(\cdot))} $ be the first eigenvalues and $ u_{n} $ be the first eigenfunctions, normalized by $ \|u_{n}\|_{\mathcal{H}_{n}} = 1 $. Under some assumptions on the variable exponents $ p_{n}(\cdot) $ and $ q_{n}(\cdot) $, we showed that $ \lambda^{1}_{(p_{n}(\cdot), \, q_{n}(\cdot))} $ converges to $ \Lambda_{\infty} $, $ u_{n} $ converges to $ u_{\infty} $ uniformly in the space $ C^{\alpha}(\Omega)\, (0 < \alpha < 1) $ and $ u_{\infty} $ is a nontrivial viscosity solution to a Dirichlet $ \infty $-Laplacian problem. Even in the case where the variable exponents reduce to the constant exponents, our work is the first one dealing with a double-phase eigenvalue problem with large exponents.