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{document}$ \begin{equation} \begin{cases} -\Delta u = f(u)&in\ \Omega\\ u = 0&on\ \partial \Omega \end{cases} \;\;\;\;\;\;\;\;(0.1)\end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^N $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ N\ge2 $\end{document}</tex-math></inline-formula> is a smooth bounded domain. Both cases where <inline-formula><tex-math id="M3">\begin{document}$ u $\end{document}</tex-math></inline-formula> is a positive or nodal solution will be considered.</p>

On the shape of solutions to elliptic equations in possibly non convex planar domains

“…It is known that the number of critical points of solutions of the previous problem is strongly influenced by the function f and by the geometry of the domain Ω. In this context the literature is too wide to report all the result, so we just focus on some of them which are closest to the interest of this paper (see [17,13] for a recent survey on the topic in the case f = f (u) and the monograph [14]). The first fundamental result has been proved for the solution of the torsion problem, i.e.…”