Vazquez in 1984 established a strong maximum principle for the classical m-Laplace differential inequality ∆ m u − f (u) ≤ 0, where ∆ m u =div(|Du| m−2 Du) and f (u) is a non-decreasing continuous function with f (0) = 0. We extend this principle to a wide class of singular inequalities involving quasilinear divergence structure elliptic operators, and also consider the converse problem of compact support solutions in exterior domains.
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
We consider ground states of the quasilinear equationthat is, C 1 solutions of (1.1) such thatTwo particular model operators A motivate this work, firstthe Laplace case, and second
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