Symmetry Results for Solutions of Semilinear Elliptic Equations with Convex Nonlinearities

Abstract

In this paper, we study the symmetry properties of the solutions of the semilinear elliptic problemwhere 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. # 2002 Elsevier Science (USA)

Some symmetric boundary value problems and non-symmetric solutions

“…Based on the result in [6] mentioned earlier, we expect that solutions of −∆u = wu 3 are symmetric with respect to one reflection. Thus, we restrict our analysis to functions U j that are even.…”

“…Here f ′ is the derivative of a function f ∈ C 2 (R). Subsequent extensions include, among other things, classes of solutions that are not necessarily positive [6,8,9,10,11]. In particular, a results in [10] implies that, if Ω is a ball or annulus, w is radially symmetric, and f ′ is convex, then any solution u of (1.1) with Morse index n or less has an axial symmetry.…”

“…In this case, we know by [10] that any index-2 solution has one reflection symmetry. As we will describe in Section 2, it is possible to find a smooth function w > 0 on the disk such that, numerically, the corresponding equation (1.1) admits an index-2 solution u whose modulus |u| has only one reflection symmetry.…”

“…It is known [10] that any such solution is invariant under a reflection symmetry of Ω. Thus, we restrict our analysis to solutions that are invariant under R y : (x, y) → (x, −y).…”

Asymptotic behaviour, nodal lines and symmetry properties for solutions of superlinear elliptic equations near an eigenvalue

“…But in such cases the authors assume to deal with positive solutions in presence of functions g's of class C 1 (for example, see [12,16,20] and the references quoted therein). For example, in [12] the authors consider the problem…”