The asymptotic behaviour of the sequence of solutions for a family of equations involving $p(\cdot)$-Laplace operators

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain with smooth boundary and let $p\colon \overline\Omega\rightarrow(1,\infty)$ be a continuous function. In this paper, we establish the existence of a positive real number $\lambda^\star$ such that for each $\lambda\in(0,\lambda^\star)$ and each integer number $n>N$ the equation $-\mathrm{div}(|\nabla u(x)|^{np(x)-2}\nabla u(x))=\lambda e^{u(x)}$ when $x\in\Omega$ subject to the homogenous Dirichlet boundary condition has a nonnegative solution, say $u_n$. Next, we prove the uniform convergence of the sequence $\{u_n\}$, as $n\rightarrow\infty$, to the distance function to the boundary of the domain $\Omega$.