Singular solutions of a Lane-Emden system
Craig Cowan, A. Razani
Abstract
<p style='text-indent:20px;'>In this work we consider the existence of positive singular solutions <p style='text-indent:20px;'><disp-formula> <tex-math id="FE11111"> \begin{document}$ \begin{equation} \left\{ \begin{array}{lcl} \hfill -\Delta u_1 & = & \lambda_1 | \nabla u_2|^p \qquad \mbox{ in } \Omega, \\ \hfill -\Delta u_2 & = & \lambda_2 | \nabla u_1|^q \qquad \mbox{ in } \Omega, \\ \hfill u_1 = u_2 & = & 0 \hfill \mbox{ on } \partial \Omega, \end{array}\right.\;\;\;\;\;\;\;(1) \end{equation} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is small <inline-formula><tex-math id="M2">\begin{document}$ C^2 $\end{document}</tex-math></inline-formula> perturbation of the unit ball <inline-formula><tex-math id="M3">\begin{document}$ B_1 $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \lambda_i $\end{document}</tex-math></inline-formula> are positive constants. Under suitable conditions on <inline-formula><tex-math id="M6">\begin{document}$ p $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ q $\end{document}</tex-math></inline-formula> we prove the existence of positive singular solutions of (1). We also examine the case where one or both of <inline-formula><tex-math id="M8">\begin{document}$ u_1,u_2 $\end{document}</tex-math></inline-formula> are Hölder continuous.