DOI: 10.68381/jca27072 ISSN: 0944-6532
Multiplicity of Positive Solutions for an Anisotropic Problem via Sub-Supersolution Method and Mountain Pass Theorem
Gelson C. G. dos Santos, Giovany Figueiredo, Julio R. S. Silva
We use the sub-supersolution method and the Mountain Pass Theorem in order to show existence and multiplicity of solution for an anisotropic problem given by
\begin{cases} \ -\Big[\displaystyle\sum^{N}_{i=1}\frac{\partial}{\partial x_{i}} \Big( \Big\vert \frac{\partial u}{\partial x_{i}}\Big\vert^{pi-2} \frac{\partial u}{\partial x_{i}}\Big )\ \Big]=a(x)u+ h(x,u) \text{ in } \Omega\text{,}\\[1mm] \ u>0\text{ in }\Omega, \quad u=0\text{ on } \partial\Omega\text{.} \end{cases}
{
−
[
∑
i
=
1
N
∂
∂
x
i
(
∣
∂
u
∂
x
i
∣
p
i
−
2
∂
u
∂
x
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)
]
=
a
(
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u
+
h
(
x
,
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in
Ω
,
u
>
0
in
Ω
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=
0
on
∂
Ω
.
We also prove the uniqueness of the solution for the linear anisotropic problem, a Comparison Principle for the anisotropic operator and a regularity result.