# American Institute of Mathematical Sciences

March  2019, 39(3): 1257-1268. doi: 10.3934/dcds.2019054

## 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

Received  January 2018 Published  December 2018

Fund Project: 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.

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.
Citation: Wenxiong Chen, Congming Li, Jiuyi Zhu. Fractional equations with indefinite nonlinearities. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1257-1268. doi: 10.3934/dcds.2019054
##### References:

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