Z. Ding, D. Costa and G. Chen, A high linking method for sign changing solutions for semilinear elliptic equations, Nonlinear Analysis, 38(1999) 151-172.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....theorems, such as, the mountain pass, various linking and saddle point theorems, focus on the existence issue. They require one to solve a two level global optimization problem and therefore are not for algorithm implementation. Inspired by the numerical work of Chio McKenna and Ding Costa Chen in [9,12], the idea of Nehari and Ding Ni to define a solution manifold and the Morse theory, we present a local minimax method (LMM) Consider a semilinear elliptic BVP as a model problem #u(x) f(x, u(x) 0 x (1.1) for u H = H where# is bounded and open, f(x, t) is nonlinear satisfying ....

Z. Ding, D. Costa and G. Chen, "A high linking method for sign changing solutions for semilinear elliptic equations", Nonlinear Analysis, 38(1999) 151-172. 9

....the existence issue. They require one to solve a two level global optimization problem and therefore are not for algorithm implementation and they can not precisely describe the local instability behavior of a saddle point. Inspired by the numerical works of Choi McKenna [8] and Ding Costa Chen [13], and motivated by the Morse theory and the idea to define a (stable) solution manifold [26] 27] 28] a local minimax method is developed in [20] which characterizes a saddle point as a local minimax solution. Based on the local minimax characterization, a numerical local minimax algorithm is ....

Z. Ding, D. Costa and G. Chen, A high linking method for sign changing solutions for semilinear elliptic equations, Nonlinear Analysis, 38 (1999) 151-172.

....[19] 20] 21] 24] such as the mountain pass, various linking and saddle point theorems, require one to solve a two level global optimization problem and therefore not useful for algorithm implementation. Efforts for numerically computing saddle points have been made in [7] for MI=i and in [10] for MI:2 which were motivated by theoretical (global minimax) characterizations of saddle points in [1] and [23] respectively. Inspired by [7, 10] and an idea in [9] a local minimax method (we shall refer it as LMM in the paper) was developed in [12, 13] and many multiple solutions were ....

....problem and therefore not useful for algorithm implementation. Efforts for numerically computing saddle points have been made in [7] for MI=i and in [10] for MI:2 which were motivated by theoretical (global minimax) characterizations of saddle points in [1] and [23] respectively. Inspired by [7, 10] and an idea in [9] a local minimax method (we shall refer it as LMM in the paper) was developed in [12, 13] and many multiple solutions were numerically computed for a class of semilinear elliptic equations. Its convergence results are obtained in [13] Several results in instability analysis of ....

Z. Ding, D. Costa and G. Chen, A high linking method for sign changing solutions for semilinear elliptic equations, Nonlinear Analysis, 38(1999) 151-172.

....problem and therefore not for algorithm implementation. On the other hand, the local theory which studies the local characterization, local behavior and local instability of critical points has not been developed. In [10] motivated by the numerical works of Choi McKenna [4] and Ding Costa Chen [6] and the idea to define a solution manifold [15, 17] a new local minimax method which characterizes a saddle point as a solution to a two level local minimax problem, is developed. The basic idea of the method is to define a local peak selection [10, 23] Let H be a Hilbert space and L C H be a ....

Z. Ding, D. Costa and G. Chen, A high linking method for sign changing solutions for semilinear elliptic equations, Nonlinear Analysis, 38(1999), 151-172.

....the existence issue. They require one to solve a two level global optimization problem and therefore are not for algorithm implementation and they can not precisely describe the local instability behavior of a saddle point. Inspired by the numerical works of Choi McKenna [10] and Ding Costa Chen [15], by the Morse theory and the idea to define a (stable) solution manifold [25] 26] 27] a local minimax method has been developed in [20] which requires one to solve only a local minimax problem. Based on the local minimax characterization, a numerical local minimax algorithm has been ....

Z. Ding, D. Costa and G. Chen, "A high linking method for sign changing solutions for semilinear elliptic equations", Nonlinear Analysis, 38(1999) 151-172.

....(See [1] 19] 20] 21] 22, 25] such as the mountain pass, various linking and saddle point theorems, require one to solve a two level global minimax problem and therefore not for algorithm implementation. In [14] motivated by the numerical works of Choi McKenna [7] and Ding Costa Chen [12], the Morse theory and the idea to define a solution submanifold, a new local minimax theorem which characterizes a saddle point as a solution to a two level local minimax problem is developed. Based on the local characterization, a new numerical minimax algorithm for finding multiple saddle ....

....solutions are invisible. Thus those examples will not be repeated here. Instead, we present numerical convergence data and profiles of multiple solutions to the Henon s equation and a sublinear elliptic equation which have some distinct features from those numerical examples previously computed in [7, 12, 6, 14]. Lemma 2.1 If p is a local peak selection of J w.r.t. L at a point v # S L # s.t. i) p is continuous at v, ii) dis(L, p(v) 0 and (iii) #J # (p(v) # 0, then s(v) 0. Proof Applying Lemma 1.1 with v # = v. Theorem 2.1 Let v k and w k = p(v k ) k=0,1,2, be the sequence ....

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Z. Ding, D. Costa and G. Chen, A high linking method for sign changing solutions for semilinear elliptic equations, Nonlinear Analysis, 38(1999) 151-172. 38

....[1] 16] 17] 18] 19, 22] such as the mountain pass, various linking and saddle point theorems, require one to solve a two level global optimization problem and therefore not for algorithm implementation. In [12] motivated by the numerical works of Choi McKenna [6] and Ding Costa Chen [11], the Morse theory and the idea to define a solution submanifold, new local minimax theorems which characterize a critical point as a solution to a two level local minimax problem are established. Based on the local characterization, a new numerical minimax method for finding multiple critical ....

Z. Ding, D. Costa and G. Chen, A high linking method for sign changing solutions for semilinear elliptic equations, Nonlinear Analysis,toappear.

....using an idea from Aubin Ekeland [4] in 1993 Choi McKenna [11] proposed a numerical minimax algorithm, called a mountain pass method, to solve the model problem basically for a solution with MI = 1. Since a flow chart of the algorithm is not provided in [11] that algorithm has been modified in [16] and further rewritten in [10] Since the function J in [11] has only one maximum along each direction, whether or not it is a local or global maximum at the first level is not a concern there. 5 The merit of this algorithm is that (a) at the first level, a maximization is taken over an a#ne ....

....A partial justification of that algorithm is also given there. A high linking theorem for the existence of a sign changing solution (MI = 2) is proved in [32] by constructing a local link at a mountain pass solution. Motivated by this idea, a numerical high linking method is proposed in [16] by Ding Costa Chen to solve the model problem for a sign changing solution (MI = 2) The basic idea is that, assume a mountain pass solution w 1 has been found, by using an ascent direction and a descent direction at w 1 , one can form a triangle as a local linking . Then one can proceed to find ....

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Z. Ding, D. Costa and G. Chen, A high linking method for sign changing solutions for semilinear elliptic equations, Nonlinear Analysis, 38(1999) 151-172.

....is well understood in the case of MIA, where the sequences of sub solutions and super solutions will sandwich a bona fide (stable) solution in the C 2,#(#32810 [1, 18] under certain relatively easily verifiable) conditions on (1. 1) However, for the various algorithmic versions of MPA and HLA ([4, 6, 7, 10] e.g. proofs of convergence to an unstable solution turn out to be much more challenging. In the original formulation of the famous Mountain Pass Lemma by Ambrosetti and Rabinowitz [2] its proof is not totally constructive. Also compounding this di#culty is an associated variational problem ....

Z. Ding, D. Costa and G. Chen, A high linking method for sign changing solutions of semilinear elliptic equations, Nonlin. Anal. 38 (1999), 151--172.

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Z. Ding, D. Costa and G. Chen, A high linking method for sign changing solutions for semilinear elliptic equations, Nonlinear Analysis, 38(1999) 151-172.

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Z. Ding, D. Costa and G. Chen, A high linking method for sign changing solutions for semilinear elliptic equations, Nonlinear Analysis, 38(1999), 151-172.

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Z. Ding, D. Costa and G. Chen, A high linking method for sign changing solutions for semilinear elliptic equations, Nonlinear Analysis, 38(1999) 151-172.

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Z. Ding, D. Costa and G. Chen, A high linking method for sign changing solutions for semilinear elliptic equations, Nonlinear Analysis,toappear.

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