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{$P_{\lambda }$} \end{equation*} where $\Omega \subset {\mathbb{R}}^N$ is a bounded domain with smooth boundary $\partial \Omega$, $1/2<s<1$, $\lambda >0$ is a real parameter, $0<q<1$, $N>2s$, $2_s^{\ast }=2N/(N-2s)$ and $$\mathcal{A}(u)=u{\mathcal{X}}_{{\sum }_D}+{\partial }_{\nu }u{\mathcal{X}}_{{\sum }_{\mathcal{N}}},{\partial }_{\nu }=\frac{\partial }{\partial \nu }.$$ Here ${\sum }_D$, ${\sum }_{\mathcal{N}}$ are smooth $(N-1)$ dimensional submanifolds of $\partial \Omega$ such that ${\sum }_D\cup {\sum }_{\mathcal{N}}=\partial \Omega$, ${\sum }_D\cap {\sum }_{\mathcal{N}}=\varnothing$ and ${\sum }_D\cap \overline{{\sum }_{\mathcal{N}}}={\tau }^{\prime }$ is a smooth $(N-2)$ dimensional submanifold of $\partial \Omega$. Within a suitable range of $\lambda$, we establish existence of at least two opposite energy solutions for \eqref{1} using the standard Nehari manifold technique.

References in corpus (1)

• Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains