Concentration results for a singularly perturbed elliptic system with variable coefficients

Bhakti Bhusan Manna; Alok Kumar Sahoo

doi:10.3934/dcds.2023033

Article Contents

2023, Volume 43, Issue 8: 2889-2922. Doi: 10.3934/dcds.2023033

This issue Previous Article Natural extensions and entropy of $ \alpha $-continued fraction expansion maps with odd partial quotients Next Article Variation of measures of maximal $ u $-entropy

Concentration results for a singularly perturbed elliptic system with variable coefficients

Bhakti Bhusan Manna^1, , and
Alok Kumar Sahoo^1,

1.
Dept. of Mathematics, IIT Hyderabad, India

^*Corresponding author: Bhakti Bhusan Manna
^*Corresponding author: Bhakti Bhusan Manna

Received: November 2022

Revised: March 2023

Early access: April 2023

Published: August 2023

The last author is supported by a Thesis fellowship from Ministry of Education, Govt. of India

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Abstract

In this article, we study the following Hamiltonian elliptic system with coefficients:

$ \begin{align} \notag \left\{\begin{aligned} -\varepsilon^2\Delta u +c(x)u = b(x) |{v}|^{q-1}v, &\, \, \text{ and } -\varepsilon^2\Delta v +c(x)v = a(x) |{u}|^{p-1}u \quad\text{in } \Omega, \\ u>0, \ v>0 \text{ in } \Omega, &\, \, \text{ and }\quad\frac{ \partial u}{ \partial\nu} = 0 = \frac{ \partial v}{ \partial\nu}\quad \text{on } \partial\Omega, \end{aligned} \right. \end{align} $

where $ \Omega $ is a smooth bounded domain in $ \mathbb{R}^n, n\ge 3 $. The coefficients $ a(x), b(x), $ and $ c(x) $ are smooth, positive, and bounded. We study the existence of point-concentrating solutions and the influence of the coefficients on their concentration profiles. As an application of this result, we establish the existence of higher-dimensional concentrating solutions for a class of elliptic systems.

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Mathematics Subject Classification: Primary: 35J47; Secondary: 35J57, 35J50.

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