Multiple normalized solutions for ( 2 , q ) -Laplacian equation problems in whole R N

Abstract

This paper considers the existence of multiple normalized solutions of the following ( 2 , q ) -Laplacian equation: { − Δ u − Δ q u = λ u + h ( ϵ x ) f ( u ) , i n R N , ∫ R N | u | 2 d x = a 2 , where 2 < q < N , ϵ > 0 , a > 0 and λ ∈ R is a Lagrange multiplier which is unknown, h is a continuous positive function and f is also continuous satisfying L 2 -subcritical growth. When ϵ is small enough, we show that the number of normalized solutions is at least the number of global maximum points of h by Ekeland's variational principle.