The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities

Abstract

We study the combined effect of concave and convex nonlinearities on the numbers of positive solutions for a fractional equation involving critical Sobolev exponents. In this paper, we concerned with the following fractional equation $ \left\{ \begin{array}{l}{\left( { - \Delta } \right)^s}u = \lambda f\left( x \right){\left| u \right|^{q - 2}}u + g\left( x \right){\left| u \right|^{2_s^* - 2}}u,\;\;\;x \in \Omega ,\\u = 0,\;\;x \in \partial \Omega ,\end{array} \right.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right) $ where $ 0<s<1 $, $ λ>0 $, $ 1≤q <2$, $ 2_s^* = \frac{2N}{N-2s} $, $ 0∈ Ω\subset \mathbb{R} ^N(N>4s) $ is a bounded domain with smooth boundary $ \partialΩ $, and $ f,\,g $ are nonnegative continuous functions on $\bar{Ω} $. Here $ (-Δ)^s $ denotes the fractional Laplace operator.