CRITICAL GROWTH PROBLEMS FOR 1/2-LAPLACIAN IN R

We study the existence of weak solutions for fractional elliptic equations of the type (-Delta)(1/2)u + V(x)u = h(u), u > o in R where h is a real valued function that behaves like e(u2) as u -> infinity and V( x) is a positive continuous unbounded function. Here (-Delta) 1 2 is the fractional Laplacian operator. We show the existence of mountain-pass solution when the nonlinearity is superlinear near t = 0. We also study the corresponding critical exponent problem for the Kirchhoff equation [GRAPHICS] where f(u) behaves like e(u2) as u -> infinity and f(u) similar to u(theta) , with theta > 3, as u -> 0