Multiple positive solutions for a critical fractional p-Laplacian system with concave nonlinearities

Abstract

The aim of this paper is to study the following nonlinear fractional p-Laplacian system with critical exponents: { ( − Δ ) p s u + | u | p − 2 u = λg ( x ) | u | q − 2 u + α α + β f ( x ) | u | α − 2 u | v | β  in  Ω , ( − Δ ) p s v + | v | p − 2 v = μh ( x ) | v | q − 2 v + β α + β f ( x ) | u | α | v | β − 2 v  in  Ω , u = v = 0  in  R N ∖Ω , where Ω is a smooth bounded set in R N , 0<s<1, 0 $ ]]> λ , μ > 0 are two parameters, 1 < q < p < p s ∗ , N>ps, 1 $ ]]> α , β > 1 satisfy α + β = p s ∗ with p s ∗ = np n − ps is the fractional Sobolev critical exponent and ( − Δ ) p s is the fractional p-Laplacian operator. Using the Nehari manifold and Ljusternik–Schnirelmann category, we study the topology of the global maximum set Θ of f ( x ) , and show that the system has at least at least ca t Θ δ ( Θ ) + 1 distinct positive solutions.