Limiting behavior and local uniqueness of normalized solutions for mass critical Kirchhoff equations

Abstract

In present paper, we study the limit behavior of normalized ground states for the following mass critical Kirchhoff equation2 and Q is the unique positive radially symmetric solution of equation −2∆uWe consider the existence of constraint minimizers for the associated energy functional involving the parameter a. The minimizer corresponds to the normalized ground state of above problem, and it exists if and only if a > 0. Moreover, when V (x) attains its flattest global minimum at an inner point or only at the boundary of Ω, we analyze the fine limit profiles of the minimizers as a ց 0, including mass concentration at an inner point or near the boundary of Ω. In particular, we further establish the local uniqueness of the minimizer if it is concentrated at a unique inner point.

Existence and Limit Behavior of Constraint Minimizers for a Varying Non-Local Kirchhoff-Type Energy Functional

“…For η = 0, s > 0, p = 4(s+1) 3 and if (V 1 ) holds, then I(η, s, λ) has at least one minimizer if 0 < λ < λ * . Moreover, I(η, s, λ) has no minimizer for λ ≥ λ * Remark that similar conclusions appear elsewhere for studying different types of Kirchhoff equations, as seen in [7,12,14,15]. For convenience, we give a detailed proof of Theorems 1 and 2 in Section 3.…”

“…In papers [10][11][12][13], the authors studied the existence and non-existence of constraint minimizers for the Kirchhoff-type energy functional with a L 2 -subcritical term. Also for V(x) being a polynomial function, the articles [14,15] obtained the limit behavior of L 2 -norm solutions when η > 0 as b → 0 + or b > 0 as η → 0 + . Coincidentally for s = 0 and R 3 replaced by R 2 , the (1) comes from an interesting physical context, which is associated with the well known Bose-Einstein condensates (BECs).…”

Local uniqueness of multi-peak positive solutions to a class of fractional Kirchhoff equations

“…We refer to to e.g. [11,13,17,18,23,26] for more mathematical researches on Kirchhoff type equations in the whole space. We also refer to [34] for a recent survey of the results connected to this model.…”

Blow up Analysis of Constraint Minimizers for a Class of Elliptic Equations with Bi-nonlocal Terms

“…In addition, for s = 1, r = 0, β > 0 and V (x) fulfilling suitable choices, (1.6) is regarded as a Kirchhoff type energy functional and there are many works related to studying the existence and limit behavior of constraint minimizers for (1.5) (cf. [14,22,23,27,31,34,41,42,43,44]).…”