Concentrated solutions for a critical elliptic equation

Abstract

<p style='text-indent:20px;'>In this paper, we are concerned with the following elliptic equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} -\Delta u = Q(x)u^{2^*-1 }+\varepsilon u^{s}, \; &amp;{\;{\rm{in}}\;\; \Omega},\\ \ u&gt;0, \; &amp;{\;{\rm{in}}\;\; \Omega}, \\ \ u = 0, &amp;{\;{\rm{on}}\;\; \partial \Omega}, \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ N\geq 3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ s\in [1, 2^*-1) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M3">\begin{document}$ 2^* = \frac{2N}{N-2} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ \varepsilon&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a smooth bounded domain in <inline-formula><tex-math id="M6">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>. Under some conditions on <inline-formula><tex-math id="M7">\begin{document}$ Q(x) $\end{document}</tex-math></inline-formula>, Cao and Zhong in Nonlin. Anal. TMA (Vol 29, 1997,461–483) proved that there exists a single-peak solution for small <inline-formula><tex-math id="M8">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> if <inline-formula><tex-math id="M9">\begin{document}$ N\geq 4 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ s\in (1, 2^*-1) $\end{document}</tex-math></inline-formula>. And they proposed in Remark 1.7 of their paper that</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{array}{c} ''it\; is\; interesting\; to\; know\; the \;existence\; of\; single-peak \;solutions \; for \;small \;\varepsilon\;\\ and \;s = 1''.\end{array} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Also it was addressed in Remark 1.8 of their paper that</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ \begin{array}{c} ''the\; question\; of \;solutions \;concentrated \;at \;several\; points \;at \;the \;same \;time \;is \\ still \;open''. \end{array}$\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Here we give some confirmative answers to the above two questions. Furthermore, we prove the local uniqueness of the multi-peak solutions. And our results show that the concentration of the solutions to above problem is delicate whether <inline-formula><tex-math id="M11">\begin{document}$ s = 1 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M12">\begin{document}$ s&gt;1 $\end{document}</tex-math></inline-formula>.</p>