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Keyword: Mixed Boundary Data (1)	2 Apr 2025	Run
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articleOpen Access
Subcritical nonlocal problems with mixed boundary conditions
Bulletin of Mathematical Sciences10 Aug 2023
Preview Abstract
By using linking and $\nabla$ -theorems in this paper we prove the existence of multiple solutions for the following nonlocal problem with mixed Dirichlet–Neumann boundary data,
$\{\begin{matrix} {(- Δ)}^{s} u = λ u + f (x, u) & in Ω, \\ u = 0 & on Σ_{𝒟}, \\ \frac{\partial u}{\partial ν} = 0 & on Σ_{𝒩}, \end{matrix}$
where ${(- Δ)}^{s}$ , $s \in (1 / 2, 1)$ , is the spectral fractional Laplacian operator, $Ω \subset ℝ^{N}$ , $N > 2 s$ , is a smooth bounded domain, $λ > 0$ is a real parameter, $ν$ is the outward normal to $\partial Ω$ , $Σ_{𝒟}$ , $Σ_{𝒩}$ are smooth $(N - 1)$ -dimensional submanifolds of $\partial Ω$ such that $Σ_{𝒟} \cup Σ_{𝒩} = \partial Ω$ , $Σ_{𝒟} \cap Σ_{𝒩} = \emptyset$ and $Σ_{𝒟} \cap {\bar{Σ}}_{𝒩} = Γ$ is a smooth $(N - 2)$ -dimensional submanifold of $\partial Ω$ .

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