The strong maximum principle revisited

Abstract

In this paper we first present the classical maximum principle due to E. Hopf, together with an extended commentary and discussion of Hopf's paper. We emphasize the comparison technique invented by Hopf to prove this principle, which has since become a main mathematical tool for the study of second order elliptic partial differential equations and has generated an enormous number of important applications. While Hopf's principle is generally understood to apply to linear equations, it is in fact also crucial in nonlinear theories, such as those under consideration here.In particular, we shall treat and discuss recent generalizations of the strong maximum principle, and also the compact support principle, for the case of singular quasilinear elliptic differential inequalities, under generally weak assumptions on the quasilinear operators and the nonlinearities involved. Our principal interest is in necessary and sufficient conditions for the validity of both principles; in exposing and simplifying earlier proofs of corresponding results; and in extending the conclusions to wider classes of singular operators than previously considered.The results have unexpected ramifications for other problems, as will develop from the exposition, e.g.(i) two point boundary value problems for singular quasilinear ordinary differential equations (Sections 3 and 4); (ii) the exterior Dirichlet boundary value problem (Section 5); (iii) the existence of dead cores and compact support solutions, i.e. dead cores at infinity (Section 7); (iv) Euler-Lagrange inequalities on a Riemannian manifold (Section 9); (v) comparison and uniqueness theorems for solutions of singular quasilinear differential inequalities (Section 10).

Classification of positive D1,p(RN)-solutions to the critical p

“…[16] recalling that the p-Laplace operator is no more degenerate in the set {∇u = 0}. By standard regularity theory, since we know that {∇u = 0} = ∅, it follows that u ∈ C 2,α loc (R N \ {0}) and (|u ′ | p−2 u ′ r N −1 ) ′ = 0.…”

Stable solutions of $−\Delta u = f(u)$ in $\mathbb{R}^N$

“…Make the same assumptions as in Theorem 1.19. Assume in addition that f satisfies the local estimates (16), (17). For p 0 in the supercritical range (15), any bounded positive solution u ∈ C 2 (R N ) of (1) which is stable outside a compact set satisfies…”

“…Assume p 0 is in the supercritical range (15) and f satisfies the local bounds (16), (17) as well as the global inequality (20). Then u ≡ 0.…”

A maximum principle for some nonlinear cooperative elliptic PDE systems with mixed boundary conditions

“…[10,18,17,19,21,22,23,24,29] and the references therein. Maximum principles often essentially express that the studied equation is a qualitatively reliable model of the underlying real phenomenon, e.g.…”