A critical fractional equation with concave–convex power nonlinearities

Abstract

In this work we study the following fractional critical problem (P_{\lambda }) = \begin{cases} (−\mathrm{\Delta })^{s}u = \lambda u^{q} + u^{2_{s}^*−1},\:u > 0 & \text{in }\Omega , \\ u = 0 & \text{in }\mathbb{R}^{n} \setminus \Omega , \end{cases} where \Omega \subset \mathbb{R}^{n} is a regular bounded domain, \lambda > 0 , 0 < s < 1 and n > 2s . Here (−\mathrm{\Delta })^{s} denotes the fractional Laplace operator defined, up to a normalization factor, by −(−\mathrm{\Delta })^{s}u(x) = \int \limits_{\mathbb{R}^{n}}\frac{u(x + y) + u(x−y)−2u(x)}{|y|^{n + 2s}}\:dy,\:x \in \mathbb{R}^{n}. Our main results show the existence and multiplicity of solutions to problem (P_{\lambda }) for different values of λ . The dependency on this parameter changes according to whether we consider the concave power case ( 0 < q < 1 ) or the convex power case ( 1 < q < 2^*_{s}−1 ). These two cases will be treated separately.