On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains: clustering concentration layers

TL;DR

This paper investigates the formation of clustering concentration layers along curves for solutions to a nonlinear elliptic PDE in two-dimensional bounded domains, considering variable coefficients and boundary conditions.

Contribution

It introduces new results on the existence and structure of clustering solutions concentrating on curves, extending previous work to variable coefficient operators and boundary effects.

Findings

Existence of solutions with clustering layers along curves.

Characterization of concentration profiles as epsilon tends to zero.

Influence of variable coefficients on the shape and location of concentration layers.

Abstract

We consider the clustering concentration on curves for solutions to the problem $$ \varepsilon^2 {\mathrm {div}}\big( \nabla_{{\mathfrak a}(y)} u\big)- V(y)u+u^p\, =\, 0, \quad u>0 \quad\mbox{in }\Omega, \qquad \nabla_{{\mathfrak a}(y)} u\cdot \nu\, =\, 0\quad\mbox{on } \partial \Omega, $$ where $\Omega$ is a bounded domain in $\mathbb R^2$ with smooth boundary, the exponent $p$ is greater than $1$, $\varepsilon>0$ is a small parameter, $V$ is a uniformly positive smooth potential on $\bar{\Omega}$, and $\nu$ denotes the outward normal of $\partial \Omega$. For two positive smooth functions ${\mathfrak a}_1(y), {\mathfrak a}_2(y)$ on $\bar\Omega$, the operator $\nabla_{{\mathfrak a}(y)}$ is given by $$ \nabla_{{\mathfrak a}(y)} u=\Bigg({\mathfrak a}_1(y)\frac{\partial u}{\partial y_1}, \, {\mathfrak a}_2(y)\frac{\partial u}{\partial y_2}\Bigg). $$