Approximate nonradial solutions for the Lane-Emden problem in the ball

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.

Keywords

• semilinear elliptic equations
• bifurcation
• sign changing solutions
• 35j15
• 35j60
• 35b06
• 35b32