  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.
