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

Abstract

In this paper, we consider the following magnetic Schrödinger equation−ΔAu+V(x)u=Enu+|u|p−2u,x∈R2, where 2<p<∞, A(x)=(bx22,−bx12), x=(x1,x2)∈R2, En is an eigenvalue of −ΔA 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∞ norms tend to zero along this sequence. To prove this, a new critical point theorem without the Palais-Smale condition is established.