DOI: 10.68381/jca21055 ISSN: 0944-6532
Multiple Ordered Positive Solutions of an Elliptic Problem Involving the p&q-Laplacian
Francisco Julio S. A. Corrêa, Amanda Suellen S. Corrêa, João R. Santos Junior
We are concerned with questions of existence and multiplicity of positive solutions of the elliptic problem
\left\{\begin{array}{rclcc} -\mathrm{div} \;(\mathcal{K}(|\nabla u|^{p})|\nabla u|^{p-2}\nabla u) & = & \lambda f(u) & \text{in} & \Omega, \\ u & = & 0 & \text{on}& \partial\Omega \end{array} \right. \tag{$P_{\lambda}$}
{
−
d
i
v
(
K
(
∣
∇
u
∣
p
)
∣
∇
u
∣
p
−
2
∇
u
)
=
λ
f
(
u
)
in
Ω
,
u
=
0
on
∂
Ω
(
P
λ
)
with
1<p<\infty
1
<
p
<
∞
, where
\Omega \subset \mathbb{R}^{N}
Ω
⊂
R
N
is a bounded smooth domain,
\lambda
λ
is a positive real parameter,
\mathcal{K}:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}
K
:
R
+
→
R
+
is a
C^{1}
C
1
-function and
f:\mathbb{R}\rightarrow \mathbb{R}
f
:
R
→
R
is a continuous functions which changes sign. We use variational methods.