Let \Omega be a bounded domain in \mathbf {R}^n, n \ge 3, with a boundary \partial \Omega \in C^2. We consider the following singularly perturbed nonlinear elliptic problem on \Omega : \varepsilon ...^2 \Delta u - u + f(u) = 0, u > 0 \textrm { on }\Omega , \quad u = 0 \textrm { on } \partial \Omega , where the nonlinearity f is of subcritical growth. Under rather strong conditions on f, it has been known that for small \varepsilon > 0, there exists a mountain pass solution u_\varepsilon of above problem which exhibits a spike layer near a maximum point of the distance function d from \partial \Omega as \varepsilon \to 0. In this paper, we construct a solution u_\varepsilon of above problem which exhibits a spike layer near a maximum point of the distance function under certain conditions on f, which we believe to be almost optimal.
For a compact smooth manifold
(
M
,
g
0
)
with a boundary, we study the conformal rigidity and non-rigidity of the scalar curvature in the conformal class. It is known that the sign of the first ...eigenvalue for a linearized operator of the scalar curvature by a conformal change determines the rigidity/non-rigidity of the scalar curvature by conformal changes when the scalar curvature
R
g
0
is positive. In this paper, we show the sign condition of
R
g
0
is not necessary, and a reversed rigidity of the scalar curvature in the conformal class does not hold if there exists a point
x
0
∈
M
with
R
g
0
(
x
0
)
>
0
.
There have been numerous studies on Hardy’s inequality on a bounded domain, which holds for functions vanishing on the boundary. On the other hand, the classical Legendre differential equation ...defined in an interval can be regarded as a Neumann version of the Hardy inequality with subcritical weight functions. In this paper we study a Neumann version of the Hardy inequality on a bounded C^2-domain in \mathbb {R}^n of the following form \begin{equation*} \int _\Omega d^{\beta }(x) |\nabla u(x) |^2 dx \ge C(\alpha ,\beta ) \int _\Omega \frac {|u(x)|^2}{d^{\alpha }(x)} dx \quad \text { with }\quad \int _\Omega \frac {u(x)}{d^{\alpha }(x)} dx=0, \end{equation*} where d(x) is the distance from x \in \Omega to the boundary \partial \Omega and \alpha ,\beta \in \mathbb {R}. We classify all (\alpha ,\beta ) \in \mathbb {R}^2 for which C(\alpha ,\beta ) > 0. Then, we study whether an optimal constant C(\alpha ,\beta ) is attained or not. Our study on C(\alpha ,\beta ) for general (\alpha ,\beta ) \in \mathbb {R}^2 shows that the (classical) Hardy inequality can be regarded as a special case of the Neumann version.
In this paper, we prove the existence of a positive least energy vector solution and its asymptotic behavior for three-component nonlinear Schrödinger systems with mixed couplings (two repulsive and ...one attractive) forces and nonconstant potentials on the entire space when the interaction forces are large. When the system with mixed coupling forces has constant potentials, repulsive forces make a state more stable when the interaction between components are less; thus it loses a compactness due to translation segregating components. To get a compactness, we impose a potential wall at infinity; then, we can construct a least energy vector solution. A main interest in this work is its asymptotic behavior of the solution for large interaction forces; one component repelling other two components survives and the other two components diminish and concentrate at a point diverging to infinity as the interaction forces are getting larger and larger. The location of the concetration point, which we could characterize in terms of the limit of a surviving component, a repulsive force and potentials of diminishing components under the assumption of the nondegeneracy for the limit problem of the surviving component.
We consider a singularly perturbed elliptic equation \\e^2\Delta u - V(x) u + f(u)=0, \ u(x) > 0 \textrm{ on } \RN, \, \lim_{|x| \to \infty}u(x) = 0,\ where $V(x) > 0$ for any $x \in \RN.$ The ...singularly perturbed problem has corresponding limiting problems \\Delta U - c U + f(U)=0, \ U(x) > 0 \textrm{ on } \ \RN, \, \lim_{|x| \to \infty}U(x) = 0, \ c > 0. \ Berestycki-Lions found almost necessary and sufficient conditions on nonlinearity $f$ for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of potential $V$ under possibly general conditions on $f$. In this paper, we prove that under the optimal conditions of Berestycki-Lions on $f \in C^1$, there exists a solution concentrating around topologically stable positive critical points of $V$, whose critical values are characterized by minimax methods.
This paper is concerned with the existence and qualitative property of standing wave solutions (ProQuest: Formulae and/or non-USASCII text omitted; see image) for the nonlinear Schrödinger equation ...(ProQuest: Formulae and/or non-USASCII text omitted; see image) with E being a critical frequency in the sense that (ProQuest: Formulae and/or non-USASCII text omitted; see image) . We show that there exists a standing wave which is trapped in a neighbourhood of isolated minimum points of V and whose amplitude goes to 0 as (ProQuest: Formulae and/or non-USASCII text omitted; see image) . Moreover, depending upon the local behaviour of the potential function V(x) near the minimum points, the limiting profile of the standing-wave solutions will be shown to exhibit quite different characteristic features. This is in striking contrast with the non-critical frequency case (ProQuest: Formulae and/or non-USASCII text omitted; see image) which has been extensively studied in recent years.PUBLICATION ABSTRACT
In this paper we study the existence of multi-bump positive solutions of the following nonlinear elliptic problem:
−
Δ
u
=
u
p
in
Ω
t
,
u
=
0
on
∂
Ω
t
.
Here
1
<
p
<
N
+
2
N
−
2
when
N
≥
3
,
1
<
p
<
...∞
when
N
=
2
and
Ω
t
is a tubular domain which expands as
t
→
∞
. See (
1.6
) below for a precise definition of expanding tubular domain. When the section
D
of
Ω
t
is a ball, the existence of multi-bump positive solutions is shown by Dancer and Yan (Commun Partial Differ Equ, 27(1–2), 23–55,
2002
) and by Ackermann et al. (Milan J Math, 79(1), 221–232,
2011
) under the assumption of a non-degeneracy of a solution of a limit problem. In this paper we introduce a new local variational method which enables us to show the existence of multi-bump positive solutions without the non-degeneracy condition for the limit problem. In particular, we can show the existence for all
N
≥
2
without the non-degeneracy condition. Moreover we can deal with more general domains, for example, a domain whose section is an annulus, for which least energy solutions of the limit problem are really degenerate.
This paper is concerned with the qualitative property of the ground state solutions for the Hénon equation. By studying a limiting equation on the upper half space R+N, we investigate the asymptotic ...energy and the asymptotic profile of the ground states for the Hénon equation. The limiting problem is related to a weighted Sobolev type inequality which we establish in this paper.
Nous nous intéresserons, dans cet article, aux propriétés qualitatives des fonctions minimisantes (ou « ground state solutions ») de l'équation d'Hénon. L'étude d'une équation limite dans le demi-espace supérieur R+N, nous renseignera sur l'énergie et les caractéristiques limites des fonctions minimisantes de l'équation d'Hénon. Notons que le problème limite est en relation avec une inégalité de Sobolev pondérée que nous établirons également.
The detection of blood at a crime scene is an important process for identification and case reconstruction. However, blood may be difficult to observe with the naked eye on dark or multi-colored ...surfaces. Acidic hydrogen peroxide (AHP) is a recently reported blood enhancement reagent that can enhance blood with high sensitivity by increasing the exposure time of the camera. However, it has never been compared to previously known techniques on dark or multi-colored surfaces. For this purpose, the method of observation/photographing (UV and IR photography), alginate casting, leuco rhodamine 6G (LR6G), and AHP were applied to bloody impression on dark or multi-colored surfaces and the results were compared. As a result, blood treated with AHP had a higher contrast to the surfaces than UV and IR photography, and it was applicable on all surfaces, opposed to alginate casting. In addition, AHP successfully enhanced blood on dark or multi-colored surfaces, similar to LR6G. 범죄 현장에서 혈액을 찾아 식별하는 것은 신원 확인 및 사건 재구성을 하기 위해 중요한 과정이다. 하지만, 혈액은 어둡거나 다양한 색상의 표면에서 육안으로 관찰하기 어려울 수 있다. Acidic hydrogen peroxide (AHP)는 최근에 발표된 혈액 증강 시약으로, 카메라의 장노출 기능을 사용하면 혈액을 높은 감도로 관찰할 수 있다. 그러나 어둡거나 다양한 색상의 표면에서 기존에 알려진 기법과 비교된 바는 없다. 이를 위해, 어둡거나 다양한 색상의 표면 8 종류에 혈흔족적을 남기고 UV나 IR을 비추면서 관찰/촬영하는 방법, alginate 전사법, leuco rhodamine 6G (LR6G), AHP를 적용하여 비교하였다. 그 결과, AHP는 UV 및 IR 촬영법보다 증강한 혈액과 표면의 contrast가 높았고, alginate 전사법과 달리 모든 표면에서 적용이 가능했다. 또한 LR6G와 마찬가지로 AHP 역시 어둡거나 다양한 색상의 표면에 부착된 혈액을 성공적으로 증강하였다.
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