A concentration phenomenon for semilinear elliptic equations

TL;DR

This paper studies how solutions to a semilinear elliptic equation concentrate at specific points as the positive region of a coefficient shrinks, revealing a concentration phenomenon in the solutions.

Contribution

It demonstrates the concentration behavior of solutions when the positive region of the coefficient shrinks to a point or two points, providing new insights into solution localization.

Findings

Solutions concentrate at the point where the positive region shrinks.

Concentration occurs in both $H^1$ and certain $L^q$-norms.

Ground state solutions concentrate at one of two points when the positive region shrinks to two points.

Abstract

For a domain $\Omega\subset\dR^N$ we consider the equation $ -\Delta u + V(x)u = Q_n(x)\abs{u}^{p-2}u$ with zero Dirichlet boundary conditions and $p\in(2,2^*)$. Here $V\ge 0$ and $Q_n$ are bounded functions that are positive in a region contained in $\Omega$ and negative outside, and such that the sets $\{Q_n>0\}$ shrink to a point $x_0\in\Omega$ as $n\to\infty$. We show that if $u_n$ is a nontrivial solution corresponding to $Q_n$, then the sequence $(u_n)$ concentrates at $x_0$ with respect to the $H^1$ and certain $L^q$-norms. We also show that if the sets $\{Q_n>0\}$ shrink to two points and $u_n$ are ground state solutions, then they concentrate at one of these points.