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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 $λ$. Moreover, the potential behaves like a potential well when the parameter λ is large. Using variational methods, combining Nehari methods, we prove that the equation has a least energy solution which, as the parameter λ large, localizes near the bottom of the potential well. Moreover, if the zero set int \begin{document} $V^{-1}(0)$ \end{document} of \begin{document} $V(x)$ \end{document} includes more than one isolated component, then \begin{document} $u_\lambda (x)$ \end{document} will be trapped around all the isolated components. However, in Laplacian case when \begin{document} $s=1$ \end{document} , for \begin{document} $\lambda$ \end{document} large, the corresponding least energy solution will be trapped around only one isolated component and will become arbitrary small in other components of int \begin{document} $V^{-1}(0)$ \end{document} . This is the essential difference with the Laplacian problems since the operator \begin{document} $(-Δ)^{s}$ \end{document} is nonlocal.

Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth