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

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. Academic Press

A Priori Estimates and Existence on Strongly Coupled Cooperative Elliptic Systems

“…In turn, u = u(x n ) depends only on x n and is non-decreasing in x n . (We refer the reader to [8] for a detailed proof.) Now we claim that κ i x …”

“…We employ a proof used in [10] and refer the reader to [10] for further details. (See also [8,11].) We shall use some notations from [10].…”

A Liouville-type theorem for the 3-dimensional parabolic Gross–Pitaevskii and related systems

“…Concerning the Liouville property, it is well known that the Liouville-type result for (4) plays an important role in the study elliptic problems as well. The optimal Liouville-type theorem for nonnegative solutions of (4) was completely proved by Reichel and Zou [32] (see also [14]) via moving sphere techniques, under the optimal Sobolev subcritical range p < p S , where p := 2r + 1 and…”

“…One of the main difficulties is that the techniques of moving planes or moving spheres do not work as in the elliptic case [32]. As for system (1) under assumption (2), only some partial cases are known:…”

A note on non-existence results for semi-linear cooperative elliptic systems via moving spheres

“…The system (1.1) was studied in [2] and several results on monotonicity, onedimensionality and non-existence on non-negative solutions were obtained. The following notations were used in [2].…”