Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains

Abstract

We study the qualitative properties of sign changing solutions of the Dirichlet problem Δ u + f ( u ) = 0 in Ω, u = 0 on ∂ Ω, where Ω is a ball or an annulus and f is a C 1 function with f ( 0 ) ⩾ 0 . We prove that any radial sign changing solution has a Morse index bigger or equal to N + 1 and give sufficient conditions for the nodal surface of a solution to intersect the boundary. In particular, we prove that any least energy nodal solution is non radial and its nodal surface touches the boundary. To cite this article: A. Aftalion, F. Pacella, C. R. Acad. Sci. Paris, Ser. I 339 (2004).