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

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|r−1u+|u|q−1u,x∈Ω⊂Rl,u=0,x∈∂Ω,where Ω is an open bounded domain, 0<r<p−1<q<p∗−1, 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.