Fractional equations with indefinite nonlinearities

Wenxiong Chen; Congming Li; Jiuyi Zhu

doi:10.3934/dcds.2019054

Article Contents

2019, Volume 39, Issue 3: 1257-1268. Doi: 10.3934/dcds.2019054

This issue Previous Article Fundamental solutions of a class of homogeneous integro-differential elliptic equations Next Article Direct methods on fractional equations

Fractional equations with indefinite nonlinearities

1.
Department of Mathematical Sciences, Yeshiva University, New York NY 10033, USA
2.
School of Mathematics, Shanghai Jiao Tong University, Shanghai, China
3.
Department of Mathematics, Louisiana State University, Baton Rouge LA 70803, USA

* Corresponding author, partially supported by NSFC 11571233
* Corresponding author, partially supported by NSFC 11571233

Received: December 31, 2017

Published: December 2018

The first author is partially supported by the Simons Foundation Collaboration Grant for Mathematicians 245486
The third author is partially supported by NSF DMS 1500468

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Abstract

In this paper, we consider a fractional equation with indefinite nonlinearities

$(-\vartriangle )^{α/2} u = a(x_1) f(u) $

for $0<α<2$, where $a$ and $f$ are nondecreasing functions. We prove that there is no positive bounded solution. In particular, in the case $a(x_1) = x_1$ and $f(u) = u^p$, this remarkably improves the result in [15] by extending the range of $α$ from $[1,2)$ to $(0,2)$, due to the introduction of new ideas, which may be applied to solve many other similar problems.

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Mathematics Subject Classification: Primary: 35J60; Secondary: 35B06.

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