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

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Abstract

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 Δpu=λ|u|r1u+|u|q1u,xΩRl,u=0,xΩ,where Ω is an open bounded domain, 0<r<p1<q<p1, λ>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.

MSC

58E05
58E30
35J92
65N12
65N30

Keywords

Ljusternik–Schnirelman critical point theory
Ljusternik–Schnirelman minimax algorithm
p-Laplacian equation
Finite element method
Convergence

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