Asymptotic Analysis

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 .

In this work, using Ljusternik–Schnirelman category theory, we study the existence and multiplicity of nontrivial solutions for the following class of elliptic Kirchhoff–Boussinesq type problems given by ε 4 Δ 2 u ± ε p Δ p u + V ( x ) u = f ( u ) + β | u | 2 * * − 2 u in  R N , u ∈ H 2 ( R N ) , where ε > 0 , 2 < p < 2 * = 2 N / ( N − 2 ) , 2 * * = 2 N / ( N − 4 ) and N ≥ 5 , V : R N → R is a continuous function and f : R → R is a function of C 1 class. We consider the subcritical case, that is, β = 0 and critical case, that is, β = 1 .

This paper is concerned with a class of abstract viscoelastic equations in which the delay element appears in the nonlinear internal damping and infinite memory is considered. We study the stability result of energy for a wide range of kernel functions and establish the decay estimates of the energy solution. The energy decay is more general and explicit under appropriate conditions for both delayed and nondelayed frictional dampings by introducing an appropriate Lyapunov function and some arguments of convex functions. Finally, a semilinear wave system as an application and some examples of energy decay rates are presented.