Concentration and multiple normalized solutions for a class of biharmonic Schrödinger equations1

In the present paper, we study the existence and concentration of multiple normalized solutions to the following nonlinear biharmonic Schrödinger equation: ε 4 Δ 2 u + V ( x ) u = λ u + h ( u ) , x ∈ R N , ∫ R N | u | 2 d x = c 2 ε N , x ∈ R N , where ε > 0 is a positive parameter, λ ∈ R is unknown and appears as a Lagrange multiplier, and V is a positive potential such that inf Λ V < inf ∂ Λ V for some open bounded subset Λ ⊂ R N ( N ⩾ 5 ). Applying the penalization techniques and Ljusternik–Schnirelmann theory, we obtain multiple mormalized solutions u ε . When ε → 0, these solutions concentrates around a local minimum of V. This paper extends the results of Alves and Thin (2023), which considered the nonlinear Schrödinger equations with general nonlinearities, to the biharmonic Schrödinger equations. We develop a truncated skill to obtain the minimum via careful analysis. Moreover, we also obtain orbital stability of the solutions.