Bifurcation Method for Solving Positive Solutions to a Class of Semilinear Elliptic Equations and Stability Analysis of Solutions
نویسندگان
چکیده
Semilinear elliptic equations are ubiquitous in natural sciences. They give rise to a variety of important phenomena in quantum mechanics, nonlinear optics, astrophysics, etc because they have rich multiple solutions. But the nontrivial solutions of semilinear equations are hard to be solved for the lack of stabilities, such as Lane-Emden equation, Henon equation and Chandrasekhar equation. In this paper, bifurcation method is applied to solving semilinear elliptic equations which are with homogeneous Dirichlet boundary conditions in 2D. Using this method, nontrivial numerical solutions will be computed and visualized in many different domains (such as square, disk, annulus, dumbbell, etc). Keywords—Semilinear elliptic equations; positive solutions; bifurcation method; isotropy subgroups.
منابع مشابه
Existence and multiplicity of positive solutions for a class of semilinear elliptic system with nonlinear boundary conditions
This study concerns the existence and multiplicity of positive weak solutions for a class of semilinear elliptic systems with nonlinear boundary conditions. Our results is depending on the local minimization method on the Nehari manifold and some variational techniques. Also, by using Mountain Pass Lemma, we establish the existence of at least one solution with positive energy.
متن کاملExistence, Uniqueness and Stability of Positive Solutions for a Class of Semilinear Elliptic Systems
We consider the stability of positive solutions to semilinear elliptic systems under a new general sublinear condition and its variants. Using the stability result and bifurcation theory, we prove the existence and uniqueness of positive solution and obtain the precise global bifurcation diagram of the system being a single monotone solution curve.
متن کاملStructure of the solution set for a class of semilinear elliptic equations with asymptotic linear nonlinearity
We consider a semilinear elliptic equation with asymptotic linear nonlinearity applying bifurcation theory and spectral analysis. We obtain the exact multiplicity of the positive solutions and a very precise structure of the solution set, which improves the previous knowledge of the problem.
متن کاملAlgorithms and Visualization for solutions of nonlinear Elliptic equations
In this paper, we compute and visualize solutions of several major types of semilinear elliptic boundary value problems with a homogeneous Dirichlet boundary condition in 2D. We present the mountain–pass algorithm (MPA), the scaling iterative algorithm (SIA), the monotone iteration and the direct iteration algorithms (MIA and DIA). Semilinear elliptic equations are well known to be rich in thei...
متن کاملAutomated bifurcation Analysis for Nonlinear Elliptic Partial Difference Equations on Graphs
We seek solutions u ∈ R to the semilinear elliptic partial difference equation −Lu+ fs(u) = 0, where L is the matrix corresponding to the Laplacian operator on a graph G and fs is a one-parameter family of nonlinear functions. This article combines the ideas introduced by the authors in two papers: a) Nonlinear Elliptic Partial Difference Equations on Graphs (J. Experimental Mathematics, 2006),...
متن کامل