Journal ArticleDOI
Conditions for two-peaked solutions of singularly perturbed elliptic equations
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In this article, it was shown that there are no two-peaked solutions of singularly perturbed elliptic equations in a strictly convex domain, and that these conditions are related to the geometry of the domain.Abstract:
We obtain necessary conditions for the existence of two-peaked solutions of singularly perturbed elliptic equations. These conditions are related to the geometry of the domain. In particular, we prove there are no two-peaked solutions in a strictly convex domain.read more
Citations
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Journal ArticleDOI
Strongly interacting bumps for the Schrödinger–Newton equations
Juncheng Wei,Matthias Winter +1 more
TL;DR: In this article, the authors studied the concentrated bound states of the Schrodinger-Newton equations and proved the existence and uniqueness of ground states of Δψ−ψ+Uψ=0, ψ>0, x∊R3; ψ(x)→0, U(x)-→0 as |x|→∞.
Journal Article
On interacting bumps of semi-classical states of nonlinear Schrödinger equations
Xiaosong Kang,Juncheng Wei +1 more
TL;DR: In this paper, concentrated positive bound states of the Schrodinger equation were studied and it was shown that at a local maximum point $x_0$ of the potential function $V(x)$ and for arbitrary positive integer $K (K>1), there always exist solutions with interacting bumps concentrating near the maximum point.
Journal ArticleDOI
Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory
TL;DR: In this article, the authors studied a perturbed semilinear problem with Neumann boundary condition and showed that for any fixed positive integer K any “suitable” critical point >>\s$(x_0^1,\dots,x_ 0^K)$>>\s tends to zero.
Book ChapterDOI
CHAPTER 3 - Qualitative Properties of Solutions to Elliptic Problems
TL;DR: In this paper, a survey of qualitative properties of elliptic solutions to elliptic equations is presented, focusing on two properties of solutions: the shape of solutions and the stability of solutions.
Journal ArticleDOI
On energy minimizers of the diblock copolymer problem
Xiaofeng Ren,Juncheng Wei +1 more
TL;DR: In this article, Nishiura and Ohnishi this article considered the long range interaction of the monomers in a chain and showed that the interfacial energy density at bonding points is proportional to the thickness of interfaces between the two monomers.
References
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Local minimizers and singular perturbations
Robert V. Kohn,Peter Sternberg +1 more
Abstract: We construct local minimisers to certain variational problems. The method is quite general and relies on the theory of Γ-convergence. The approach is demonstrated through the model problem It is shown that in certain nonconvex domains Ω ⊂ ℝ n and for e small, there exist nonconstant local minimisers u e satisfying u e ≈ ± 1 except in a thin transition layer. The location of the layer is determined through the requirement that in the limit u e → u 0 , the hypersurface separating the states u 0 = 1 and u 0 = −1 locally minimises surface area. Generalisations are discussed with, for example, vector-valued u and “anisotropic” perturbations replacing |∇u| 2 .
Journal ArticleDOI
Boundary and interior transition layer phenomena for pairs of second-order differential equations☆
Paul C. Fife,Paul C. Fife +1 more
TL;DR: In this article, the authors considered a family of solutions with a limit in some sense as l + 0, where l is the number of vertices in the solution and E is the length of the transition from the limit to the boundary.
Journal ArticleDOI
On the Construction of Single-Peaked Solutions to a Singularly Perturbed Semilinear Dirichlet Problem
TL;DR: In this article, the problem of finding spiky solutions of (1.1) has been studied in various applications, such as chemotaxis, population genetics, and chemical reactor theory.
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