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September  2020, 19(9): 4349-4362. doi: 10.3934/cpaa.2020196

## Asymptotic behavior of solutions for nonlinear integral equations with Hénon type on the unit Ball

 Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China

* Corresponding author

Received  September 2019 Revised  March 2020 Published  June 2020

Fund Project: The corresponding author is supported by the National Natural Sciences Foundations of China(No:11671128)

In this paper, we consider the problem
 $\begin{equation*} f^{q-1}(x) = \int_{\Omega}\frac{|x|^{\alpha}|y|^{\beta}f(y)}{|x-y|^{n-\gamma}}dy, \; f>0, \; x\in\overline{\Omega}, \end{equation*}$
where
 $\Omega$
is the unit ball in
 $\mathbb{R}^n(n\geq3)$
centered at the origin,
 $1<\gamma and $ \alpha, \beta>0 $, $ q_\gamma: = \frac{2n}{n+\gamma}
. We will investigate the asymptotic behavior of energy maximizing positive solution as
 $q\rightarrow (\frac{2n}{n+\gamma})^{+} = (q_\gamma)^+$
. We also show that the energy maximizing positive solution concentrate at a point, which is located at the boundary as
 $q\rightarrow (q_\gamma)^{+}$
. In addition, the energy maximizing positive solution is non-radial provided that
 $q$
closes to
 $q_\gamma$
.
Citation: Ziyi Cai, Haiyang He. Asymptotic behavior of solutions for nonlinear integral equations with Hénon type on the unit Ball. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4349-4362. doi: 10.3934/cpaa.2020196
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