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

Miaomiao Niu; Zhongwei Tang

doi:10.3934/dcds.2017168

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2017, Volume 37, Issue 7: 3963-3987. Doi: 10.3934/dcds.2017168

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

Miaomiao Niu and
Zhongwei Tang

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

Received: September 30, 2015

Revised: February 28, 2017

Published: August 2017

The second author was supported by National Science Foundation of China(11571040)

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35J65.

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