Non-Existence Results for Semilinear Cooperative Elliptic Systems via Moving Spheres

By Wolfgang Reichel, Henghui Zou. 2000

162 citations recorded by the source corpus.

Abstract Based upon a new development of the method of moving spheres, we introduce a new and general approach for non-existence of positive solutions of cooperative semilinear elliptic systems with the Laplacian as principal part. For supercritical nonlinearities we prove non-existence on bounded star-shaped domains. For subcritical nonlinearities we obtain non-existence results on a class of unbounded domains, which includes e.g. the entire space, certain curved halfspaces and the complement of bounded star-shaped domains. As a by-product we also get a symmetry result on halfspaces.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/f7ef470e62b7ac7295207afc4c7c41aca3277ca7

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Towards a Unified Approach to Nonexistence of Solutions for a Class of Differential Inequalities (2004, background): “See for instance [34] and the references therein for elliptic problems, [13], [14] ,[21],[33], and [1] for nonlinear parabolic and hyperbolic equations respectively.”
A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system (2010, background): “3) have been studied extensively in recent years, see [5,6,19,22,23,25,31] and the references therein.”
A priori Estimates and Existence for Elliptic Systems via Bootstrap in Weighted Lebesgue Spaces (2004, background): “the comments after Theorem 4 in [29]).”
A priori estimates for a semilinear elliptic system without variational structure and their applications (2002, background): “Theorem 5.6 (Reichel and Zou [ 20 ]) Let n ≥ 3 and let Ω ⊂ R n be a bounded”
Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems (2007, methodology): “3): for example, when both p and q are subcritical [13, 25] (i.”
Lane–Emden systems with negative exponents (2010, unknown): “, [1, 3, 6, 7, 16, 18, 19, 20, 21, 22, 23] and the references therein).”
Nonexistence results for a class of fractional elliptic boundary value problems (2012, methodology): “The approach we follow here is inspired by Reichel and Zou [27] who used the technique of moving spheres to prove nonexistence results for cooperative elliptic systems.”