A Numerical Algorithm for Positive Solutions of Concave and Convex Elliptic Equation on R2

Authors: Hailong Zhu, Zhaoxiang Li

Abstract:

In this paper we investigate numerically positive solutions of the equation -Δu = λuq+up with Dirichlet boundary condition in a boundary domain ╬® for λ > 0 and 0 < q < 1 < p < 2*, we will compute and visualize the range of λ, this problem achieves a numerical solution.

Keywords: positive solutions, concave-convex, sub-super solution method, pseudo arclength method.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1074597

References:

[1] R. Fowler, Further studies of Emden-s and similar differential equations, Quart. J. Math.1931,2:259-288.
[2] I. Flores, A resonance phenomenon for ground states of an elliptic equation of Emden-Fowler type,J. Differential Equations. 2004,198:1-15.
[3] R. Kajikiya, Non-radial solutions with group invariance for the sublinear Emden-Fowler equation, Nonlinear Analysis. 2001,47:3759-3770.
[4] Tsung-fang Wu. Four positive solutions for a semilinear elliptic equation involving concave and convex nonlinearities, Nonlinear Analysis.2009,70:1377-1392.
[5] Kuan-Ju Chen. Combined effects of concave and convex nonlinearities in elliptic equation on RN, J.Math.Anal.Appl.2009, 355:767-777.
[6] A. Ambrosetti. Combined effects of concave and convex nonlinearities in some elliptic problems, J.Funct.Anal. 1994,122:519-543.
[7] R. Kress. Numerical Analysis, Berlin: Springer-Verlag, 2003, p.103.
[8] R. Seydel. Practical Bifurcation and Stability Analysis, Berlin: Springer- Verlag. 1994, p.135.