Existence and uniqueness of constraint minimizers for the planar Schrödinger-Poisson system with logarithmic potentials

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

“…Comment that the blow up behavior of constraint minimizers in our paper is quite difference from these conclusions in [15,17,18,19,39]. Although the sinusoidal potential sin 2 |x| may attain its minimum at an inner point or some boundary point of Ω, one can rule out the case of minimizers blow up near the boundary.…”

“…In particular, we notice that for s, r = 0, p = 2 and β > 0 in (1.6), it is a hot research topic related to the well-known Gross-Pitaevskii functional (see [10,13]), which is derived from physical experimental phenomena of Bose-Einstein condensates. Roughly speaking, when the external potential V (x) in (1.3) behaves like polynomial, logarithmic, ring-shaped, multi-well and periodic, in these papers the authors have established some results of constraint minimizers on the existence, mass concentration behavior and local uniqueness under L 2 -critical state (see [15,16,17,18,19,24,30,39]). 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.…”

“…We care about what happens to the constraint minimizers for any β with β ↗ β * , and hence the refined energy estimation of I(s, p, r, β) as β ↗ β * is necessary. In effect, one knows from [15,17,18,19,39] that when potential V (x) behaves different forms, the key steps in estimating energy are to deal with the term V (x)u 2 dx. However, since our elliptic equations (1.4) not only contains bi-nonlocal terms, but potential V (x) is a sinusoidal function, the techniques in [17,18,19,39] are ineffective for dealing with our problem.…”

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

“…More precisely, when the external trapping potentials V(x) are in the forms of polynomial, ring-shaped, multi-well, periodic and sinusoidal, the articles [18][19][20][21][22] gave the existence, non-existence and mass concentration behavior analysis of the ground states. If V(x) behaves like logarithmic or homogeneous potential [23,24], the local uniqueness and refined spike profiles of ground states for the GP functional are analyzed when λ tends to a critical value λ * .…”

Blow-Up Analysis of L2-Norm Solutions for an Elliptic Equation with a Varying Nonlocal Term

“…Examples for s = 0, p = 2, and λ > 0 in (1), and F(u) in ( 3) are called Gross-Pitaevskii [14,15] energy functionals, which are associated with the attractive Bose-Einstein condensates [16,17]. Many mathematicians are devoted to studying the existence, non-existence, mass concentration behavior, and local uniqueness of solutions when trapping potentials take the forms of polynomial, ring-shaped, multi-well, periodic, and sinusoidal functions; see [18][19][20][21][22][23][24]. Also, for n ≤ 4, s = 1, and λ > 0, (2) is related to a Kirchhoff-type constraint minimization problem, which has attracted considerable attention from researchers analyzing the existence, local uniqueness, and blow-up behavior of minimizers.…”