A double-phase eigenvalue problem with large exponents

In the present article, we consider a double-phase eigenvalue problem with large exponents. Let λ ( p n , q n ) 1 {\lambda }_{\left({p}_{n},{q}_{n})}^{1} be the first eigenvalues and u n {u}_{n} be the first eigenfunctions, normalized by ‖ u n ‖ ℋ n = 1 \Vert {u}_{n}{\Vert }_{{{\mathcal{ {\mathcal H} }}}_{n}}=1 . Under some assumptions on the exponents p n {p}_{n} and q n {q}_{n} , we show that λ ( p n , q n ) 1 {\lambda }_{\left({p}_{n},{q}_{n})}^{1} converges to Λ ∞ {\Lambda }_{\infty } and u n {u}_{n} converges to u ∞ {u}_{\infty } uniformly in the space C α ( Ω ) {C}^{\alpha }\left(\Omega ) , and u ∞ {u}_{\infty } is a nontrivial viscosity solution to a Dirichlet ∞ \infty -Laplacian problem.