Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium
Let \((M^n,g)\), \(n\geq 3\), be a smooth compact Riemannian manifold, and let \(A: M\to M^s_p(\mathbb{R})\), \(p\in\mathbb{N}\), be a \(C^1\)-map into the space of symmetric \((p\times p)\)-matrices. The paper is concerned with the strongly coupled system \[ \Delta_g\mathcal{U}+A(x)\mathcal{U}=|\mathcal{U}|^{2^*-2}\mathcal{U} \tag{A} \] for \(\mathcal{U}=(u_1, \dots, u_p):M\to\mathbb{R}^p\); here \(\Delta_g\) is the Laplace-Beltrami operator on \(M\) acting on each component of \(\mathcal{U}\), \(|\mathcal{U}|=\left(\sum^p_{i=1}u_i^2\right)^{1/2}\), and \(2^*=2n/(n-2)\) is the critical Sobolev exponent. The system is said to be analytically (resp., weakly) stable if, given a sequence \(A_\alpha\to A\) in \(C^1(M,M^s_p(\mathbb{R}))\), and given a sequence of nonnegative, nontrivial solutions \(\mathcal{U}_\alpha\) of (A\(_\alpha\)), then, up to a subsequence, \(\mathcal{U}_\alpha\to\mathcal{U}\) in \(C^2\)(resp., weakly in \(H^1\)), and \(\mathcal{U}\) is a nonnegative nontrivial solution of (A). In dimensions \(n=4\), \(n\neq 6\), the authors present conditions on \(A\) guaranteeing that (A) is analytically stable. In dimension \(n=6\), the conditions imply only that the system is weakly stable. The result is sharp: the conditions on \(A\) cannot be dropped, and an example shows that analytic stability need not hold in dimension \(n=6\). Dimension \(n=3\) is dealt with in another paper of the authors and \textit{J. Vétois} [J. Funct. Anal. 258, No.~3, 999--1059 (2010; Zbl 1183.58018)]. The authors also discuss related concepts like compactness and geometric stability for (A). Proofs are based on an analysis of the blow-up of sequences of solutions, and on the classification of nonnegative \(H^1\)-solutions of the limit system \(-\Delta \mathcal{U}=|\mathcal{U}|^{2^*-2}\mathcal{U}\) in \(\mathbb{R}^n\).
scientific article; zbMATH DE number 5706101