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On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains: clustering concentration layers
Suting Wei, Jun Yang
We consider the clustering concentration on curves for solutions to the problem $$ \varepsilon^2 {\mathrm {div}}\big( \nabla_{{\mathfrak a}(y)} u\big)- V(y)u+u^p\, =\, 0, \quad u>0…
Clustering of Boundary Interfaces for an inhomogeneous Allen-Cahn equation on a smooth bounded domain
Lipeng Duan, Suting Wei, Jun Yang
We consider the inhomogeneous Allen-Cahn equation $$ ε^2Δu\,+\,V(y)(1-u^2)\,u\,=\,0\quad \mbox{in}\ Ω, \qquad \frac {\partial u}{\partial ν}\,=\,0\quad \mbox{on}\ \partial Ω, $$ wh…
On the non-degeneracy of radial vortex solutions for a coupled Ginzburg-Landau system
Lipeng Duan, Jun Yang
For the following Ginzburg-Landau system in \begin{align*} \begin{cases} -Δw^+ +\Big[A_+\big(|w^+|^2-{t^+}^2\big)+B\big(|w^-|^2-{t^-}^2\big)\Big]w^+=0, \\[3mm] -Δw^…
On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains
Suting Wei, Bin Xu, Jun Yang
We consider the problem $$ ε^2 Δu-V(y)u+u^p\,=\,0,~~u>0~~\quad\mbox{in}\quadΩ,~~\quad\frac {\partial u}{\partial ν}\,=\,0\quad\mbox{on}~~~\partial Ω, $$ where is a bounded doma…
Concentration on Surfaces for a Singularly Perturbed Neumann Problem in Three-Dimensional Domains
Ying Guo, Jun Yang
We consider the following singularly perturbed elliptic problem $$ \varepsilon^2\triangle\tilde{u}-\tilde{u}+\tilde{u}^p=0, \ \tilde{u}>0\quad \mbox{in} \ Ω,\ \ \ \frac{\partial\ti…