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 ${\lambda }_{(p_n(\cdot ),q_n(\cdot ))}^1$ be the first eigenvalues and $u_n$ be the first eigenfunctions, normalized by $\Vert u_n{\Vert }_{{\mathcal{H}}_n}=1$. Under some assumptions on the variable exponents $p_n(\cdot )$ and $q_n(\cdot )$, we showed that ${\lambda }_{(p_n(\cdot ),q_n(\cdot ))}^1$ converges to ${\mathrm{\Lambda }}_{\mathrm{\infty }}$, $u_n$ converges to $u_{\mathrm{\infty }}$ uniformly in the space $C^{\alpha }(\mathrm{\Omega })(0<\alpha <1)$ and $u_{\mathrm{\infty }}$ is a nontrivial viscosity solution to a Dirichlet $\mathrm{\infty }$-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.