Stable solutions of semilinear elliptic problems in convex domains

• X. Cabré ^[1] ; S. Chanillo ^[2]
  1. [1] Pierre and Marie Curie University

     Pierre and Marie Curie University

     París, Francia

     
  2. [2] Rutgers University

     Rutgers University

     City of New Brunswick, Estados Unidos

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 4, Nº. 1, 1998, págs. 1-10
• Idioma: inglés
• DOI: 10.1007/s000290050022
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• Resumen

  • In this note, we consider semilinear equations −Δu=f(u) , with zero Dirichlet boundary condition, for smooth and nonnegative f, in smooth, bounded, strictly convex domains of RN . We study positive classical solutions that are semi-stable. A solution u is said to be semi-stable if the linearized operator at u is nonnegative definite. We show that in dimension two, any positive semi-stable solution has a unique, nondegenerate, critical point. This point is necessarily the maximum of u. As a consequence, all level curves of u are simple, smooth and closed. Moreover, the nondegeneracy of the critical point implies that the level curves are strictly convex in a neighborhood of the maximum of u. Some extensions of this result to higher dimensions are also discussed.