Symmetry Results for Solutions of Semilinear Elliptic Equations with Convex Nonlinearities

✍ Scribed by Filomena Pacella

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 135 KB
• Volume: 192
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2001.3901

✦ Synopsis

In this paper, we study the symmetry properties of the solutions of the semilinear elliptic problem

( where O is a bounded symmetric domain in R N , N 52, and f : O Â R ! R is a continuous function of class C 1 in the second variable, g is continuous and f and g are somehow symmetric in x. Our main result is to show that all solutions of the above problem of index one are axially symmetric when O is an annulus or a ball, g 0 and f is strictly convex in the second variable. To do this, we prove that the nonnegativity of the first eigenvalue of the linearized operator in the caps determined by the symmetry of O is a sufficient condition for the symmetry of the solution, when f is a convex function.