Multiplicity of normalized solutions to biharmonic Schrödinger equation with mixed nonlinearities

Abstract

This paper is concerned with the existence of multiple normalized solutions to the elliptic problems { Δ 2 u = λu + h ( ϵx ) | u | q − 2 u + | u | p − 2 u in R N , ∫ R N u 2 d x = c , where c, 0 $ ]]> ϵ > 0 , N ≥ 5 , 2 < q < 2 + 8 N < p ≤ 4 ∗ := 2 N N − 4 , λ ∈ R is a Lagrangian multiplier and h : R N → [ 0 , ∞ ) is a continuous function. We find some 0 $ ]]> c 0 > 0 and present a class of reasonable assumptions on h to guarantee that the numbers of normalized solutions are at least the numbers of global maximus points of h when ε is small enough and 0 < c ≤ c 0 .