Affiliation:
1. School of Mathematics and Statistics, North China University of Water Resources and Electric Power , Zhengzhou , 450046 , PR China
Abstract
Abstract
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.
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