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} (-\Delta)^su=\lambda u^q+u^{2_s^*-1},\ u>0\quad\text{in }\Omega,\\[3pt] \mkern+51mu u=0\quad\text{on } \Sigma_{\mathcal{D}}\\ \mkern+36mu \displaystyle \frac{\partial u}{\partial \nu}=0\quad\text{on } \Sigma_{\mathcal{N}} \end{array} \right. \end{equation*} where $\Omega\subset\mathbb{R}^N$ 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, $\lambda>0$, $\nu$ is the outwards normal to $\partial\Omega$, $\Sigma_{\mathcal{D}}$, $\Sigma_{\mathcal{N}}$ are smooth $(N-1)$-dimensional submanifolds of $\partial\Omega$ such that $\Sigma_{\mathcal{D}}\cup\Sigma_{\mathcal{N}}=\partial\Omega$, $\Sigma_{\mathcal{D}}\cap\Sigma_{\mathcal{N}}=\emptyset$, and $\Sigma_{\mathcal{D}}\cap\overline{\Sigma}_{\mathcal{N}}=\Gamma$ is a smooth $(N-2)$-dimensional submanifold of $\partial\Omega$.\newline In particular, we will prove that, for the sublinear case $0<q<1$, there exists at least two solutions for every $0<\lambda<\Lambda$ for certain $\Lambda\in\mathbb{R}$ while, for the superlinear case $1<q<2_s^*-1$, we will prove that there exists at least one solution for every $\lambda>0$. We will also prove that solutions are bounded.