A concentration phenomenon for semilinear elliptic equations

Abstract

For a domain $Ω\subset\dR^N$ we consider the equation $ -Δu + V(x)u = Q_n(x)\abs{u}^{p-2}u$ with zero Dirichlet boundary conditions and $p\in (2,2^{\ast })$. 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.

References in corpus (3)

• Solitons in nonlinear lattices
• Solitons supported by localized nonlinearities in periodic media
• Multibump nodal solutions for an indefinite superlinear elliptic problem

Cited by in corpus (1)

• Existence of standing waves solution for a Nonlinear Schrödinger equations in ${\mathbb{R}}^N$