DOI: 10.68381/jca22020 ISSN: 0944-6532
On a Nonlocal Multivalued Problem in an Orlicz-Sobolev Space via Krasnoselskii's Genus
Giovany M. Figueiredo, Jefferson A. Santos
This paper is concerned with the multiplicity of nontrivial solutions in an Orlicz-Sobolev space for a nonlocal problem involving N-functions and theory of locally Lispchitz continuous functionals. More precisely, in this paper, we study a result of multiplicity to the following multivalued elliptic problem:
\left \{ \begin{array}{l} -M\left(\displaystyle\int_\Omega \Phi(\mid\nabla u\mid)dx\right) div\big(\phi(\mid\nabla u\mid)\nabla u\big) -\phi(|u|)u\in \partial F(u) \ \text{in}\ \Omega,\\[6mm] u\in W_0^1L_\Phi(\Omega), \end{array} \right.
{
−
M
(
∫
Ω
Φ
(
∣
∇
u
∣
)
d
x
)
d
i
v
(
ϕ
(
∣
∇
u
∣
)
∇
u
)
−
ϕ
(
∣
u
∣
)
u
∈
∂
F
(
u
)
in
Ω
,
u
∈
W
0
1
L
Φ
(
Ω
)
,
where
\Omega\subset\mathbb{R}^{N}
Ω
⊂
R
N
is a bounded smooth domain,
N\geq 2
N
≥
2
,
M
M
is continuous function,
\Phi
Φ
is an N-function with
\Phi(t)=\displaystyle\int^{|t|}_{0}\phi(s)s \ ds
Φ
(
t
)
=
∫
0
∣
t
∣
ϕ
(
s
)
s
d
s
and
\partial F(t)
∂
F
(
t
)
is a generalized gradient of
F(t)
F
(
t
)
. We use genus theory to obtain the main result