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

Abstract

In this paper, we consider the problem \begin{document}$ \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*} $\end{document} where \begin{document}$ \Omega $\end{document} is the unit ball in \begin{document}$ \mathbb{R}^n(n\geq3) $\end{document} centered at the origin, \begin{document}$ 1 and \begin{document}$ \alpha, \beta>0 $\end{document} , \begin{document}$ q_\gamma: = \frac{2n}{n+\gamma} . We will investigate the asymptotic behavior of energy maximizing positive solution as \begin{document}$ q\rightarrow (\frac{2n}{n+\gamma})^{+} = (q_\gamma)^+ $\end{document} . We also show that the energy maximizing positive solution concentrate at a point, which is located at the boundary as \begin{document}$ q\rightarrow (q_\gamma)^{+} $\end{document} . In addition, the energy maximizing positive solution is non-radial provided that \begin{document}$ q $\end{document} closes to \begin{document}$ q_\gamma $\end{document} .