LEAST ENERGY SOLUTIONS FOR NONLINEAR SCHRODINGER EQUATION INVOLVING THE FRACTIONAL LAPLACIAN AND CRITICAL GROWTH

In this paper, we study a class of nonlinear Schrodinger equations involving the fractional Laplacian and the nonlinearity term with critical Sobolev exponent. We assume that the potential of the equations includes a parameter A. Moreover, the potential behaves like a potential well when the parameter A is large. Using variational methods, combining Nehari methods, we prove that the equation has a least energy solution which, as the parameter A large, localizes near the bottom of the potential well. Moreover, if the zero set int V-1(0) of V(x) includes more than one isolated component, then u(lambda)(x) will be trapped around all the isolated components. However, in Laplacian case when s = 1, for lambda large, the corresponding least energy solution will be trapped around only one isolated component and will become arbitrary small in other components of int V-1(0). This is the essential difference with the Laplacian problems since the operator (-Delta)(s) is nonlocal.