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

Abstract

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 $\Omega \subset {\mathbb{R}}^N$ is a smooth bounded domain, $\frac{1}{2}<s<1$, $0<q<2_s^{\ast }-1$, $q\ne 1$, being $2_s^{\ast }=\frac{2N}{N-2s}$ 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 ${\Sigma }_{\mathcal{D}}\cup {\Sigma }_{\mathcal{N}}=\partial \Omega$, ${\Sigma }_{\mathcal{D}}\cap {\Sigma }_{\mathcal{N}}=\varnothing$, 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 $1<q<2_s^{\ast }-1$, we will prove that there exists at least one solution for every $\lambda >0$. We will also prove that solutions are bounded.