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
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
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:
Keywords: Variable exponent spaces, asymptotic behaviour, Ekeland’s variational principle, distance function to the boundary, viscosity solution.
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