Nonlocal critical exponent singular problems under mixed Dirichlet-Neumann boundary conditions

Abstract

In this paper, we study the following singular problem, under mixed Dirichlet-Neumann boundary conditions, and involving the fractional Laplacian \begin{equation*} \label{1} \begin{cases} (-Δ)^{s}u = λu^{-q} + u^{2^*_s-1}, \quad u>0 \quad \text{in }Ω, \mathcal A(u) = 0 \quad \text{on}~ \partialΩ= \sum_{D} \cup \sum_{\mathcal{N}}, \end{cases} \tag{$P_λ$} \end{equation*} where $Ω\subset \mathbb{R}^N$ is a bounded domain with smooth boundary $\partialΩ$, $1/2 0$ is a real parameter, $ 0 < q < 1 $, $N>2s$, $2^*_s=2N/(N-2s)$ and $$\mathcal{A}(u)= u \mathcal{X}_{\sum_{D}} + {\partial_νu}\mathcal{X}_{ \sum_{\mathcal{N}}}, \quad{\partial_ν=\frac{\partial }{\partialν}}.$$ Here $\sum_{D}$, $\sum_{\mathcal{N}}$ are smooth $(N-1)$ dimensional submanifolds of $\partial Ω$ such that $\sum_{D} \cup \sum_{\mathcal{N}}= \partialΩ$, $\sum_{D} \cap \sum_{\mathcal{N}}= \emptyset $ and $\sum_{D} \cap \overline{\sum_{\mathcal{N}}} = τ'$ is a smooth $(N-2)$ dimensional submanifold of $\partialΩ$. Within a suitable range of $λ$, we establish existence of at least two opposite energy solutions for \eqref{1} using the standard Nehari manifold technique.