Sign-changing Solutions and Phase Separation for an Elliptic System with Critical Exponent
Journal
Communications in Partial Differential Equations
Journal Volume
39
Journal Issue
10
Pages
1827-1859
Date Issued
2014
Author(s)
Abstract
We study the following elliptic system with critical exponent:(Formula presented.) Here, Ω is a smooth bounded domain of ℝN(N ≥ 6), is the critical Sobolev exponent, 0 < λ1, λ2 < λ1(Ω) and μ1, μ2 > 0, where λ1(Ω) is the first eigenvalue of - Δ with the Dirichlet boundary condition. When β = 0, this turn to be the well-known Brézis-Nirenberg problem. We show that, for each fixed β <0, this system has a sign-changing solution in the following sense: one component changes sign and has exactly two nodal domains, while the other one is positive. We also study the asymptotic behavior of these solutions as β → - ∞ and phase separation appears. Precisely, two components of these solutions tend to repel each other and converge to solutions of the Brézis-Nirenberg problem in segregated regions. © 2014 Copyright Taylor & Francis Group, LLC.
Type
journal article