DOI: 10.68381/jca33008 ISSN: 0944-6532
Spectral Study about Biharmonic Operator with Landesman-Lazer Condition
Giovany M. Figueiredo, Segundo M. A. Salirrosas, Lorena Soriano
We study the existence of weak solutions for the following class of problems
\left\{ \begin{array}{lcl} \alpha \Delta^{2}u +\beta \Delta u = \mu u +\gamma h(x,u)&\text{in}\ \Omega,\\[1mm] B(u) = 0 &\text{on}\ \partial\Omega, \end{array} \right.
{
α
Δ
2
u
+
β
Δ
u
=
μ
u
+
γ
h
(
x
,
u
)
in
Ω
,
B
(
u
)
=
0
on
∂
Ω
,
where
\Omega\subset\mathbb{R}^N
Ω
⊂
R
N
is a bounded smooth domain,
N\geq 1
N
≥
1
,
\alpha\geq0
α
≥
0
,
-\infty <\beta<\alpha\lambda_1
−
∞
<
β
<
α
λ
1
,
\lambda_1
λ
1
is the first eigenvalue of
(-\Delta, H^1_0(\Omega))
(
−
Δ
,
H
0
1
(
Ω
)
)
,
\mu\in(0,\bar{\mu})
μ
∈
(
0
,
μ
ˉ
)
,
\bar{\mu}<\mu_2
μ
ˉ
<
μ
2
,
\mu_2
μ
2
is the second eigenvalue of the problem
(\alpha\Delta^2u+\beta\Delta u,H^1_0(\Omega)\cap H^2(\Omega)),
(
α
Δ
2
u
+
β
Δ
u
,
H
0
1
(
Ω
)
∩
H
2
(
Ω
)
)
,
\gamma\neq0
γ
≠
0
is a real parameter and
h:\overline{\Omega}\times\mathbb{R} \to \mathbb{R}
h
:
Ω
‾
×
R
→
R
is a Carathéodory function verifying some conditions, the boundary condition
B(u)=0
B
(
u
)
=
0
on
\partial \Omega
∂
Ω
means that
u=\Delta u=0
u
=
Δ
u
=
0
on
\partial\Omega
∂
Ω
when
\alpha>0
α
>
0
and
u=0
u
=
0
on
\partial \Omega
∂
Ω
when
\alpha=0
α
=
0
. In this article, we revisit the arguments of Landesman-Lazer, both in the global and local aspects.