A Strong Maximum Principle for some quasilinear elliptic equations

Abstract

Abstract.In its simplest form the Strong Maximum Principle says that a nonnegative superharmonic continuous function in a domain f~ c R n, n >/1, is in fact positive everywhere. Here we prove that the same conclusion is true for the weak solutions of -Au +/3(u) = f with/3 a nondecreasing function R ~R, /3(0)=0, and f>~0 a.e. in ~ if and only if the integral f(/3(s)s)-l/2dsdiverges at = 0+. We extend the result to general S more equations, in particular to -Ap u +/3 (u) = f where Ap (u) = div( I Du ]P -2Du), 1 < p < ~. Our main result characterizes the nonexistence of a dead core in some reaction-diffusion systems.

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

“…This allows to exploit the strong maximum principle (see [24]) and get that u ′ (r) = 0 for r > 0. In particular we have that…”

Multiple Solutions for Nonlinear Coercive Problems with a Nonhomogeneous Differential Operator and a Nonsmooth Potential

“…For easy reference, we state in detail the hypotheses: Remark 3.1 Clearly, hypotheses H 0 are a particular case of hypotheses H 0 . The reason we have dropped the z-dependence is that we need an extension of the nonlinear strong maximal principle of Vázquez [35], valid for the p-Laplacian, to more general nonhomogeneous differential operators, like the one in this paper. The only such result for z-dependent operators is that of Zhang [36], who though requires that η = 0 in hypotheses H 0 (ii) and (iii).…”

Multiple solutions for resonant nonlinear periodic equations

“…Invoking the nonlinear maximum principle of Vazquez [21], we conclude that u 0 ∈ intĈ + . In a similar fashion, working this time with the functional ϕ ε − , we obtain v 0 ∈ −intĈ + , a second constant sign solution of (1.1).…”