Two classes of Ljusternik-Schnirelman minimax algorithms and an application for finding multiple negative energy solutions of a class of p-Laplacian equations

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:342

Article

Yao, Xudong¹ 

[1] Shanghai Normal Univ, Dept Math, Shanghai 200234, Peoples R China

关键词: Ljusternik Schnirelman critical point theory;    Ljusternik Schnirelman minimax algorithm;    p-Laplacian equation;    Finite element method;    Convergence;   

DOI  :  10.1016/j.cam.2018.04.020

来源: Elsevier

【 摘 要 】

In this paper, two classes of LS-minimax algorithms are presented, they are applied to numerically find multiple negative energy solutions of the p-Laplacian equation {-Delta p(u) = lambda vertical bar u vertical bar(r - 1)u + vertical bar u vertical bar(q - 1)u, x is an element of Omega subset of R-1, u = 0, x is an element of partial derivative Omega, where Omega is an open bounded domain, 0 < r < p - 1 < q < p* 1, lambda > 0, and p* is the Sobolev exponent, and mathematical justification and global convergence result for them are established. By combining LS-minimax algorithm with the finite element method, it is verified that, as element size goes to zero, numerical solutions of p-Laplacian equation captured by LS-minimax algorithm converge to solutions of p-Laplacian equation. Two LS-minimax algorithms developed in [1] are two special algorithms in these two classes of algorithms. (C) 2018 Elsevier B.V. All rights reserved.

【 授权许可】

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