A new critical point theorem and small magnitude solutions of magnetic Schrodinger equations with Landau levels

In this paper, we consider the following magnetic Schrodinger equation -Delta(A)u + V(x)u = E(n)u + |u|(p-2)u, x is an element of R-2, where 2 < p < infinity, A(x) = (bx(2)/2,-bx(1)/2 ), x = (x(1), x(2)) is an element of R-2, E-n is an eigenvalue of -Delta A with infinitely multiplicity, and V is a non-zero and nonnegative function in Lp/(p-2)(R2, R). We prove that this equation has a sequence of non-zero solutions whose L infinity norms tend to zero along this sequence. To prove this, a new critical point theorem without the Palais-Smale condition is established. (c) 2021 Elsevier Inc. All rights reserved.

Automated summary

This paper investigates the properties of solutions of the magnetic Schrodinger equation and proves the existence of a sequence of non-zero solutions whose L infinity norms tend to zero along the sequence. Additionally, a new critical point theorem without the Palais-Smale condition is established.