Critical points of solutions to elliptic problems in planar domains

Pages: 327 - 338, Volume 10, Issue 1, January 2011

doi:10.3934/cpaa.2011.10.327       Abstract        References        Full Text (871.9K)       Related Articles

Jaime Arango - Departamento de Matemáticas, Universidad del Valle, Cali, Colombia (email)
Adriana Gómez - Departamento de Matemáticas, Universidad del Valle, Cali, Colombia (email)

Abstract: Given a planar domain $\Omega$, and an analytic function $f$, we describe the set of critical points for the solution $u$ of the semilinear elliptic problem $\Delta u = f(u)$ in $\Omega$, $u=0$ on $\partial\Omega$. For simply connected domains we establish that the set of critical points is finite while for non--simply connected domains we show that this set is made up of finitely many isolated points and finitely many analytic Jordan curves. Further results are given in the case that $\Omega$ is an annular domain whose border has nonzero curvature.

Keywords:  Morse theory, critical points, semilinear Dirichlet problems, simply onnected domains.
Mathematics Subject Classification:  Primary: 74K15; Secondary: 35J05, 65M06.

Received: April 2009;      Revised: October 2009;      Published: November 2010.

 References