Blowing-up solutions for a supercritical elliptic equation

Abstract

<p style='text-indent:20px;'>This paper concerns the existence of solutions of the following supercritical PDE: <inline-formula><tex-math id="M1">\begin{document}$ (P_\varepsilon) $\end{document}</tex-math></inline-formula>: <inline-formula><tex-math id="M2">\begin{document}$ -\Delta u = K|u|^{\frac{4}{n-2}+\varepsilon}u\; \mbox{ in }\Omega, \; u = 0 \mbox{ on }\partial\Omega, $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a smooth bounded domain in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ n\geq 3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ K $\end{document}</tex-math></inline-formula> is a <inline-formula><tex-math id="M7">\begin{document}$ C^3 $\end{document}</tex-math></inline-formula> positive function and <inline-formula><tex-math id="M8">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> is a small positive real. Our method is inspired from the work of Bahri-Li-Rey. It consists to reduce the existence of a critical point to a finite dimensional system. Using a fixed-point theorem, we are able to construct positive solutions of <inline-formula><tex-math id="M9">\begin{document}$ (P_\varepsilon) $\end{document}</tex-math></inline-formula> having the form of two bubbles with non comparable speeds and which have only one blow-up point in <inline-formula><tex-math id="M10">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>. That means that this blow-up point is non simple. This represents a new phenomenon compared with the subcritical case.</p>