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.
Categories: math.AP