On a class of non-local elliptic equations with asymptotically linear term

Yuanhong Wei; Xifeng Su

doi:10.3934/dcds.2018154

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2018, Volume 38, Issue 12: 6287-6304. Doi: 10.3934/dcds.2018154

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On a class of non-local elliptic equations with asymptotically linear term

Yuanhong Wei^1, and
Xifeng Su^2, ,

1.
School of Mathematics, Jilin University, Changchun 130012, China
2.
School of Mathematical Sciences, Beijing Normal University, No. 19 XinJieKouWai St., HaiDian District, Beijing 100875, China

* Corresponding author: Xifeng Su
* Corresponding author: Xifeng Su

Received: August 31, 2017

Revised: October 31, 2017

Published: November 2018

Y. Wei is supported by National Natural Science Foundation of China (Grant No. 11301209) and Outstanding Youth Foundation of Jilin Province of China (Grant No. 20170520056JH), X. Su is supported by National Natural Science Foundation of China (Grant No. 11301513) and "the Fundamental Research Funds for the Central Universities".

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Abstract

We consider the nonlinear elliptic PDE driven by the fractional Laplacian with asymptotically linear term. Some results regarding existence and multiplicity of non-trivial solutions are obtained. More precisely, information about multiple non-trivial solutions is given under some hypotheses of asymptotically linear condition; non-local elliptic equations with combined nonlinearities are also studied, and some results of local existence and global existence are obtained. Finally, an $L^{∞}$ regularity result is also given in the appendix, using the De Giorgi-Stampacchia iteration method.

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Mathematics Subject Classification: Primary: 35A15, 35R11; Secondary: 35J67.

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