Abstract
Abstract In this paper we provide a numerical approximation of bifurcation branches from nodal radial solutions of the Lane Emden Dirichlet problem in the unit ball in ℝ2, as the exponent of the nonlinearity varies. We consider solutions with two or three nodal regions. In the first case our numerical results complement the analytical ones recently obtained in [11]. In the case of solutions with three nodal regions, for which no analytical results are available, our analysis gives numerical evidence of the existence of bifurcation branches. We also compute additional approximations indicating presence of an unexpected branch of solutions with six nodal regions. In all cases the numerical results allow to formulate interesting conjectures.
Highlights
Consider the classical Lane-Emden problem−Δu = |u|p−1u in B (1.1)u=0 on ∂B, when B ⊂ R2 is the open unit ball centered at the origin and p > 1.It is well known that (1.1) admits only one positive solution and infinitely many sign changing solutions as it can be shown by exploiting the oddness of the nonlinearity and standard variational methods.Among the sign changing solutions there are infinitely many radially symmetric ones which are obtained by applying variational methods in the space H01,rad(B) which is the subspace of H01(B) given by radial functions
In this paper we provide a numerical approximation of bifurcation branches from nodal radial solutions of the Lane Emden Dirichlet problem in the unit ball in R2, as the exponent of the nonlinearity varies
It could be proved that, for any m ∈ N, m > 1 there exists a unique radial solution to (1.1) with m nodal regions and which is positive at the origin ([13], [14], [15])
Summary
U=0 on ∂B, when B ⊂ R2 is the open unit ball centered at the origin and p > 1. The change of Morse index m(u2,p), as p varies from 1 to ∞, allows to claim that the solutions u2,p become degenerate (i.e., the linearized operator at u2,p admits zero as an eigenvalue) and this, in turn, allows to prove that some branches of nonradial solutions bifurcate from the solutions u2,p, for some values of the exponent p 5) the nonexistence of secondary bifurcations, at least up to certain values of p, 6) the bifurcation diagram in the cases 1) and 4), 7) an independent new branch of approximations with 6 nodal regions, which near p = 1 have a similar shape as the “obvious solutions” obtained by odd reflection of the unique positive solution on the spherical sector with opening angle π 5.
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