Fractional elliptic equations with critical exponential nonlinearity

Abstract

Abstract We study the existence of positive solutions for fractional elliptic equations of the type (-Δ) 1/2 u = h ( u ), u > 0 in (-1,1), u = 0 in ℝ∖(-1,1) where h is a real valued function that behaves like e u 2 as u → ∞ . Here (-Δ) 1/2 is the fractional Laplacian operator. We show the existence of mountain-pass solution when the nonlinearity is superlinear near t = 0. In case h is concave near t = 0, we show the existence of multiple solutions for suitable range of λ by analyzing the fibering maps and the corresponding Nehari manifold.