We investigate nodal radial solutions to semilinear problems of typewhere Ω is a bounded radially symmetric domain of R N (N ≥ 2) and f is a real function. We characterize both the Morse index and the degeneracy in terms of a singular one dimensional eigenvalue problem, and describe the symmetries of the eigenfunctions. Next we use this characterization to give a lower bound for the Morse index; in such a way we give an alternative proof of an already known estimate for the autonomous problem and we furnish a new estimate for Hénon type problems with f (|x|, u) = |x| α f (u). Concerning the real Hénon problem, f (|x|, u) = |x| α |u| p−1 u, we prove radial nondegeneracy and show that the radial Morse index is equal to the number of nodal zones.
In this paper we consider the problemFrom the characterization of the solutions of the linearized operator, we deduce the existence of nonradial solutions which bifurcate from the radial one when α is an even integer.
In this paper we consider the problem {-Delta = u(p) + lambda u in A, u > 0 in A, u = 0 on partial derivative A, where A is an annulus of R(N), N >= 2 and p > 1. We prove bifurcation of nonradial solutions from the radial solution in correspondence of a sequence of exponents {p(k)} and for expanding annuli
In this paper we prove symmetry results for classical solutions of semilinear elliptic equations in the whole R(N) or in the exterior of a ball, N >= 2, in the case when the nonlinearity is either convex or has a convex first derivative. More precisely we prove that solutions having Morse index j <= N are foliated Schwarz symmetric, i.e. they are axially symmetric with respect to an axis passing through the origin and nonincreasing in the polar angle from this axis. From this we deduce some nonexistence results for positive or sign changing solutions in the case when the nonlinearity does not depend explicitly on the space variable. (C) 2009 Elsevier Masson SAS. All rights reserved
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