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On finding solutions of a Kirchhoff type problem


Authors: Yisheng Huang, Zeng Liu and Yuanze Wu
Journal: Proc. Amer. Math. Soc. 144 (2016), 3019-3033
MSC (2010): Primary 35B09, 35B33, 35J15, 35J60
DOI: https://doi.org/10.1090/proc/12946
Published electronically: November 20, 2015
MathSciNet review: 3487233
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Abstract: Consider the Kirchhoff type problem

$\displaystyle (\mathcal {P})\qquad \quad \left \{\aligned-\bigg (a+b\int _{\mat... ...ad \text {on }\partial \mathbb {B}_R, \endaligned \right .\qquad \qquad \qquad $

where $ \mathbb{B}_R\subset \mathbb{R}^N(N\geq 3)$ is a ball, $ 2\leq q<p\leq 2^*:=\frac {2N}{N-2}$ and $ a$, $ b$, $ \lambda $, $ \mu $ are positive parameters. By introducing some new ideas and using the well-known results of the problem $ (\mathcal {P})$ in the cases of $ a=\mu =1$ and $ b=0$, we obtain some special kinds of solutions to $ (\mathcal {P})$ for all $ N\geq 3$ with precise expressions on the parameters $ a$, $ b$, $ \lambda $, $ \mu $, which reveals some new phenomenons of the solutions to the problem $ (\mathcal {P})$. It is also worth pointing out that it seems to be the first time that the solutions of $ (\mathcal {P})$ can be expressed precisely on the parameters $ a$, $ b$, $ \lambda $, $ \mu $, and our results in dimension four also give a partial answer to Naimen's open problems [J. Differential Equations 257 (2014), 1168-1193]. Furthermore, our results in dimension four seem to be almost ``optimal''.

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Additional Information

Yisheng Huang
Affiliation: Department of Mathematics, Soochow University, Suzhou 215006, People’s Republic of China
Email: yishengh@suda.edu.cn

Zeng Liu
Affiliation: Department of Mathematics, Suzhou University of Science and Technology, Suzhou 215009, People’s Republic of China
Email: luckliuz@163.com

Yuanze Wu
Affiliation: College of Sciences, China University of Mining and Technology, Xuzhou 221116, People’s Republic of China
Email: wuyz850306@cumt.edu.cn

DOI: https://doi.org/10.1090/proc/12946
Keywords: Kirchhoff type problem, critical Sobolev exponent, positive solution
Received by editor(s): July 20, 2015
Received by editor(s) in revised form: September 2, 2015
Published electronically: November 20, 2015
Additional Notes: The third author is the corresponding author
Communicated by: Joachim Krieger
Article copyright: © Copyright 2015 American Mathematical Society