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The limiting behavior of solutions to p-Laplacian problems with convection and exponential terms
Authors
Anderson L. A. de Araujo
Grey Ercole
Julio C. Lanazca Vargas
Publication date
27 April 2023
Publisher
View
on
arXiv
Abstract
We consider, for
a
,
l
≥
1
,
a,l\geq1,
a
,
l
≥
1
,
b
,
s
,
α
>
0
,
b,s,\alpha>0,
b
,
s
,
α
>
0
,
and
p
>
q
≥
1
,
p>q\geq1,
p
>
q
≥
1
,
the homogeneous Dirichlet problem for the equation
−
Δ
p
u
=
λ
u
q
−
1
+
β
u
a
−
1
∣
∇
u
∣
b
+
m
t
l
−
1
e
α
t
s
-\Delta_{p}u=\lambda u^{q-1}+\beta u^{a-1}\left\vert \nabla u\right\vert ^{b}+mt^{l-1}e^{\alpha t^{s}}
−
Δ
p
​
u
=
λ
u
q
−
1
+
β
u
a
−
1
∣
∇
u
∣
b
+
m
t
l
−
1
e
α
t
s
in a smooth bounded domain
Ω
⊂
R
N
.
\Omega\subset\mathbb{R}^{N}.
Ω
⊂
R
N
.
We prove that under certain setting of the parameters
λ
,
\lambda,
λ
,
β
\beta
β
and
m
m
m
the problem admits at least one positive solution. Using this result we prove that if
λ
,
β
>
0
\lambda,\beta>0
λ
,
β
>
0
are arbitrarily fixed and
m
m
m
is sufficiently small, then the problem has a positive solution
u
p
,
u_{p},
u
p
​
,
for all
p
p
p
sufficiently large. In addition, we show that
u
p
u_{p}
u
p
​
converges uniformly to the distance function to the boundary of
Ω
,
\Omega,
Ω
,
as
p
→
∞
.
p\rightarrow\infty.
p
→
∞.
This convergence result is new for nonlinearities involving a convection term.Comment: 18 page
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oai:arXiv.org:2303.00140
Last time updated on 20/03/2023