Normalized Solutions for the Choquard Equation With Potential: Concentration and Multiplicity

This article focuses on the study of multiplicity and concentration behavior of normalized solutions for a Choquard equation with a local perturbation { − Δ p u + V ( ϵ x ) | u | p − 2 u = λ | u | p − 2 u + ( I α * | u | q ) | u | q − 2 u + μ | u | s − 2 u in R N , ∫ R N | u | p d x = a p > 0 , where a , ϵ > 0 , 2 ≤ p < N , ( p ( N + α ) ) / 2 N < q < ( p 2 + p ( N + α ) ) / 2 N , p < s < p + ( p 2 / N ) , μ > 0 and λ ∈ R is an unknown parameter that appears as a Lagrange multiplier. Under natural hypotheses, combining the minimization techniques and Ljusternik–Schnirelmann category theory, we obtain the existence and concentration property of normalized solutions for ϵ > 0 sufficiently small, as well as the multiplicity result depending on the topology of the set M where the potential V attains its global minimum, which indicates that the numbers of normalized solutions is determined by the topological structure of the set M .