On the number of critical points of solutions of semilinear elliptic equations

Abstract

<p style='text-indent:20px;'>In this survey we discuss old and new results on the number of critical points of solutions of the problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE0.1"> $ \begin{equation} \begin{cases} -\Delta u = f(u)&in\ \Omega\\ u = 0&on\ \partial \Omega \end{cases} \;\;\;\;\;\;\;\;(0.1)\end{equation} $ </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">$ \Omega\subset \mathbb{R}^N $</tex-math></inline-formula> with <inline-formula><tex-math id="M2">$ N\ge2 $</tex-math></inline-formula> is a smooth bounded domain. Both cases where <inline-formula><tex-math id="M3">$ u $</tex-math></inline-formula> is a positive or nodal solution will be considered.</p>