LAUSR

Abstract In this paper, we consider the following magnetic Schrodinger equation − Δ A u + V ( x ) u = E n u + | u | p − 2 u , x ∈ R 2 , where 2 p ∞ , A ( x ) = ( b x 2 2 , − b x 1 2 ) , x = ( x 1 , x 2 ) ∈ R 2 , E n is an eigenvalue of − Δ A with infinitely multiplicity, and V is a non-zero and nonnegative function in L p / ( p − 2 ) ( R 2 , 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.