Bifurcation Method For Computing The Multiple Solutions To Boundary Value Problems Of Nonlinear Elliptic Equations

In this dissertation, We consider the boundary value problem(BVP) of nonlinear partial dif-ferential equationwhere ? is a bounded open domain in R2,?pu = div(| u|p?2 u), r≥0,?? is the boundaryof ?,x = (x1,x2),x0 is a center of ?.When p=2, the equation of problem (0-1) is called Henon equation. Three algorithms basedon the bifurcation method are applied to solving this problem.The first algorithm: We embed (0-1) to the nonlinear bifurcation problems with parameter.According to the bifurcation theory, it has nontrivial solution branches bifurcated from the trivialsolution near the bifurcation points. Along the nontrivial solution branches we can get the differ-ent symmetry solutions to problem (0-1) by the continuation method when the parameter goes to0.The second algorithm: By the Liapunov-Schmidt reduction, we can get the approximate ex-pression of the bifurcation equation of problem(0-1), from which we find the initial guess for theNewton iteration directly and solve directly problem(0-1) by the Newton method.The third algorithm: Taking r as a parameter, the symmetric positive solution to problem (0-1) when r=0 is used as a starting point on the symmetric positive solution branch to problem (0-1)with varied r which can be computed effectively by the r continuation and the Newton iterationmethod. While r is continued, the eigenvalues of Jacobian of problem (0-1) are monitored. Theeigenvalue with small absolute value are found for r. There are the potential symmetry-breakingbifurcation point. The symmetry-breaking bifurcation point is found via the extended systems.The other symmetric positive solutions are computed by the branch switching method based onthe Liapunov-Schmidt reduction. When there are simple folds on the branch of the symmetricpositive solutions, we successfully continue the solution branches through the simple folds via thepseudo-arclength algorithm.When p=2, the equation of problem (0-1)is called the p-Laplacian-Henon equation. We putforward three homotopy continuation algorithms to solve the problem (0-1).The first algorithm: p continuation. Taking the solutions to the problem (0-1) when p=2 asstarting point , we continue p to p = p?, where p? is the p of the problem (0-1).The second algorithm: homotopy continuation. We embed (0-1) to the nonlinear bifurcation problems with parameter t of the following form:Taking the solutions to Henon equation while t=1 as the initial point, we can get the different sym-metry solutions to p-Laplacian-Henon equation by the continuation method when the parameter tgoes to 0.The third algorithm: For any given p, the symmetric positive solution to problem (0-1) whenr=0 is used as a starting point on the symmetric positive solution branch to problem (0-1) with var-ied r which can be computed effectively by the r continuation and the Newton iteration method.While r is continued, the eigenvalues of Jacobian of problem (0-1) are monitored. The eigenvaluewith small absolute value are found for r. There are the potential symmetry-breaking bifurca-tion point. The symmetry-breaking bifurcation point is found via the extended systems. Theother symmetric positive solutions are computed by the branch switching method based on theLiapunov-Schmidt reduction.