Critical growth fractional elliptic systems with exponential nonlinearity. (May 2016) - Giacomoni, J.; Mishra, Pawan Kumar; Sreenadh, K.
Abstract: We study the existence of positive solutions for the system of fractional elliptic equations of the type, ( − Δ ) 1 2 u = p p + q λ f ( x ) ∣ u ∣ p − 2 u ∣ v ∣ q + h 1 ( u, v ) e u 2 + v 2, in ( − 1, 1 ), ( − Δ ) 1 2 v = q p + q λ f ( x ) ∣ u ∣ p ∣ v ∣ q − 2 v + h 2 ( u, v ) e u 2 + v 2, in ( − 1, 1 ), u, v > 0 in  ( − 1, 1 ), u = v = 0 in R ∖ ( − 1, 1 ) where 1 < p + q < 2, h 1 ( u, v ) = ( α + 2 u 2 ) ∣ u ∣ α − 2 u ∣ v ∣ β, h 2 ( u, v ) = ( β + 2 v 2 ) ∣ u ∣ α ∣ v ∣ β − 2 v and α + β > 2 . Here ( − Δ ) 1 2 is the fractional Laplacian operator. We show the existence of multiple solutions for suitable range of λ by analyzing the fibering maps and the corresponding Nehari manifold. We also study the existence of positive solutions for a superlinear system with critical growth exponential nonlinearity.