Liouville results and asymptotics of solutions of a quasilinear elliptic equation with supercritical source gradient term

2020/08/24 by Bidaut-Veron, Marie-Françoise · 3 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2008.10220

Abstract

We consider the elliptic quasilinear equation --Δ m u = u p |∇u| q in R N with q ≥ m and p > 0, 1 < m < N. Our main result is a Liouville-type property, namely, all the positive C 1 solutions in R N are constant. We also give their asymptotic behaviour : all the solutions in an exterior domain R N \B r0 are bounded. The solutions in B r0 0 can be extended as a continuous functions in B r0. The solutions in R N 0 has a finite limit l ≥ 0 as |x| → ∞. Our main argument is a Bernstein estimate of the gradient of a power of the solution, combined with a precise Osserman's type estimate for the equation satisfied by the gradient.

Cited by

• Classification of positive solutions to a class of Laplace equations with a gradient term
• Liouville results for (p,q)-Laplacian elliptic equations with source terms involving gradient nonlinearities
• Classification of solutions of an elliptic Hamilton-Jacobi equation

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