Least energy solutions of nonlinear Schrödinger equations involving the half Laplacian and potential wells

Miaomiao  Niu; Zhongwei Tang

doi:10.3934/cpaa.2016.15.1215

Article Contents

2016, Volume 15, Issue 4: 1215-1231. Doi: 10.3934/cpaa.2016.15.1215

This issue Previous Article Transition fronts in nonlocal Fisher-KPP equations in time heterogeneous media Next Article Persistence of the hyperbolic lower dimensional non-twist invariant torus in a class of Hamiltonian systems

Least energy solutions of nonlinear Schrödinger equations involving the half Laplacian and potential wells

Miaomiao Niu^1, and
Zhongwei Tang^1,

1.
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875

Received: June 2015

Revised: January 2016

Published: July 2016

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Abstract

In this paper, we are concerned with the existence of least energy solutions of nonlinear Schrödinger equations involving the half Laplacian \begin{eqnarray} (-\Delta)^{1/2}u(x)+\lambda V(x)u(x)=u(x)^{p-1}, u(x)\geq 0, \quad x\in R^N, \end{eqnarray} for sufficiently large $\lambda$, $2 < p < \frac{2N}{N-1}$ for $N \geq 2$. $V(x)$ is a real continuous function on $R^N$. Using variational methods we prove the existence of least energy solution $u(x)$ which localize near the potential well int$(V^{-1}(0))$ for $\lambda$ large. Moreover, if the zero sets int$(V^{-1}(0))$ of $V(x)$ include more than one isolated components, then $u_\lambda(x)$ will be trapped around all the isolated components. However, in Laplacian case, when the parameter $\lambda$ large, the corresponding least energy solution will be trapped around only one isolated component and become arbitrary small in other components of int$(V^{-1}(0))$. This is the essential difference with the Laplacian problems since the operator $(-\Delta)^{1/2}$ is nonlocal.

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

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