DOI: 10.1515/acv-2025-0114 ISSN: 1864-8258
Nonlocal critical problems with mixed boundary conditions and nearly resonant perturbations
Eduardo Colorado, Giovanni Molica Bisci, Alejandro Ortega, Luca Vilasi Abstract
We consider the following nonlocal critical problem with mixed Dirichlet–Neumann boundary conditions:
{
(
-
Δ
)
s
u
=
λ
u
+
|
u
|
2
s
*
-
2
u
in
Ω
,
u
=
0
on
Σ
𝒟
,
∂
u
∂
ν
=
0
on
Σ
𝒩
,
see text
\left\{\begin{aligned} \displaystyle{}(-\Delta)^{s}u&\displaystyle=\lambda u+|%
u|^{2_{s}^{*}-2}u&&\displaystyle\phantom{}\text{in ${\Omega}$,}\\
\displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on ${\Sigma_{%
\mathcal{D}}}$,}\\
\displaystyle\frac{\partial u}{\partial\nu}&\displaystyle=0&&\displaystyle%
\phantom{}\text{on ${\Sigma_{\mathcal{N}}}$,}\end{aligned}\right.
where
(
-
Δ
)
s
{(-\Delta)^{s}}
,
s
∈
(
1
2
,
1
)
{s\in(\frac{1}{2},1)}
, is the spectral fractional Laplacian operator,
Ω
⊂
ℝ
N
{\Omega\subset\mathbb{R}^{N}}
,
N
>
2
s
{N>2s}
, is a smooth bounded domain,
2
s
*
=
2
N
N
-
2
s
{2_{s}^{*}=\frac{2N}{N-2s}}
denotes the critical fractional Sobolev exponent,
λ
>
0
{\lambda>0}
is a real parameter, ν is the outward normal to
∂
Ω
{\partial\Omega}
,
Σ
𝒟
{\Sigma_{\mathcal{D}}}
,
Σ
𝒩
{\Sigma_{\mathcal{N}}}
are smooth
(
N
-
1
)
{(N-1)}
-dimensional submanifolds of
∂
Ω
{\partial\Omega}
such that
Σ
𝒟
∪
Σ
𝒩
=
∂
Ω
{\Sigma_{\mathcal{D}}\cup\Sigma_{\mathcal{N}}=\partial\Omega}
,
Σ
𝒟
∩
Σ
𝒩
=
∅
{\Sigma_{\mathcal{D}}\cap\Sigma_{\mathcal{N}}=\emptyset}
and
Σ
𝒟
∩
Σ
¯
𝒩
=
Γ
{\Sigma_{\mathcal{D}}\cap\overline{\Sigma}_{\mathcal{N}}=\Gamma}
is a smooth
(
N
-
2
)
{(N-2)}
-dimensional submanifold of
∂
Ω
{\partial\Omega}
.
By employing a
∇
{\nabla}
-theorem we prove the existence of multiple solutions when the parameter λ is in a left neighborhood of a given eigenvalue of
(
-
Δ
)
s
{(-\Delta)^{s}}
.