Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth
Miaomiao Niu Zhongwei Tang
Discrete & Continuous Dynamical Systems - A 2017, 37(7): 3963-3987 doi: 10.3934/dcds.2017168

In this paper, we study a class of nonlinear Schrödinger 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 $V^{-1}(0)$ 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 $(-Δ)^{s}$ is nonlocal.

keywords: Nonlinear Schrödinger equation least energy solution critical growth fractional Laplacian

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