Hopf maximum principle violation for elliptic equations with\n non-Lipschitz nonlinearity

2009/01/27 by Yavdat Il’yasov, Il'yasov, Yavdat, Youri V. Egorov +1
Computer Science · Mathematics · Engineering · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.0901.4191

Abstract

We consider elliptic equations with non-Lipschitz nonlinearity -
Delta u =\n
lambda |u|
^beta-1u-|u|
^alpha-1u in a smooth bounded domain \Ω\n\⊂ \ℝⁿ, n\≥ 3, with Dirichlet boundary conditions; here\n0<\α<\β<1. We prove the existence of a weak nonnegative solution which\ndoes not satisfy the Hopf boundary maximum principle, provided that \λ\nis large enough and n>2(1+\α) (1+\β)/(1-\α)(1-\β).\n

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