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Concave-Convex critical problems for the spectral fractional Laplacian with mixed boundary conditions

In this work we study the existence of solutions to the following critical fractional problem with concave-convex nonlinearities, \begin{equation*} \left \{ \begin{array}{l} (-Δ)^su=λu^q+u^{2_s^*-1},\ u>0\quad\text{in }Ω,\\[3pt] \mkern+51mu u=0\quad\text{on } Σ_{\mathcal{D}}\\ \mkern+36mu \displaystyle \frac{\partial u}{\partial ν}=0\quad\text{on } Σ_{\mathcal{N}} \end{array} \right. \end{equation*} where $Ω\subset\mathbb{R}^N$ is a smooth bounded domain, $\frac{1}{2}<s<1$, $0<q<2_s^*-1$, $q\neq 1$, being $2_s^*=\frac{2N}{N-2s}$ the critical fractional Sobolev exponent, $λ>0$, $ν$ is the outwards normal to $\partialΩ$, $Σ_{\mathcal{D}}$, $Σ_{\mathcal{N}}$ are smooth $(N-1)$-dimensional submanifolds of $\partialΩ$ such that $Σ_{\mathcal{D}}\cupΣ_{\mathcal{N}}=\partialΩ$, $Σ_{\mathcal{D}}\capΣ_{\mathcal{N}}=\emptyset$, and $Σ_{\mathcal{D}}\cap\overlineΣ_{\mathcal{N}}=Γ$ is a smooth $(N-2)$-dimensional submanifold of $\partialΩ$.\newline In particular, we will prove that, for the sublinear case $0<q<1$, there exists at least two solutions for every $0<λ<Λ$ for certain $Λ\in\mathbb{R}$ while, for the superlinear case $1<q<2_s^*-1$, we will prove that there exists at least one solution for every $λ>0$. We will also prove that solutions are bounded.