Abstract
In this paper, we study the existence of multiple solutions for the
singular problem
{
a
(
x
,
u
,
∇
u
)
-
div
(
b
(
x
,
u
,
∇
u
)
)
=
u
-
α
+
λ
c
(
x
,
u
)
in
Ω
,
u
>
0
in
Ω
,
u
=
0
on
ℝ
n
∖
Ω
,
\left\{\begin{aligned} \displaystyle{}a(x,u,\nabla u)-{\rm div}(b(x,u,\nabla u%
))&\displaystyle=u^{-\alpha}+\lambda c(x,u)&&\displaystyle\phantom{}\text{in }%
\Omega,\\
\displaystyle u&\displaystyle>0&&\displaystyle\phantom{}\text{in }\Omega,\\
\displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }{\mathbb{R}}%
^{n}\setminus\Omega,\end{aligned}\right.
where
Ω
⊂
ℝ
n
{\Omega\subset\mathbb{R}^{n}}
(
n
≥
3
)
{(n\geq 3)}
is a bounded domain
with
C
1
{C^{1}}
boundary, λ is a positive parameter,
0
<
α
≤
1
<
p
≤
n
{0<\alpha\leq 1<p\leq n}
and
p
*
=
n
p
n
-
p
{p^{*}=\frac{np}{n-p}}
is the critical exponent for Sobolev
embedding. Using the fibering maps and the Nehari manifold, we prove the
existence of at least two positive solutions for all values of the
parameter λ belonging to an open bounded interval of
ℝ
+
{\mathbb{R}_{+}}
.
Search from 250 M+ research papersSearch from 250 M+ research papers
Open Access
Just Published
May 1, 2024
Riccardo Durastanti + 1