# Moscow Mathematical Journal

Volume 20, Issue 3, July–September 2020 pp. 495–509.

The Asymptotic Behaviour of the Sequence of Solutions for a Family of Equations Involving $p(\cdot)$-Laplace Operators

**Authors**:
Maria Fărcăşeanu (1) and Mihai Mihăilescu (2)

**Author institution:**(1) Research group of the project PN-III-P4-ID-PCE-2016-0035,
“Simion Stoilow” Institute of Mathematics of the Romanian Academy,
010702 Bucharest, Romania

(2) Research group of the project PN-III-P4-ID-PCE-2016-0035,
“Simion Stoilow” Institute of Mathematics of the Romanian Academy,
010702 Bucharest, Romania;

Department of Mathematics, University of Craiova, 200585 Craiova, Romania

**Summary: **

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 $\Omega$.

2010 Math. Subj. Class. 35D40, 35J20, 46E30, 46E35, 47J20

**Keywords:**Variable exponent spaces, asymptotic behaviour, Ekelandâ€™s variational principle, distance function to the boundary, viscosity solution.

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