Liouville Results and Asymptotics of Solutions of a Quasilinear Elliptic Equation with Supercritical Source Gradient Term

We consider the elliptic quasilinear equation -Δm⁢u=up⁢|∇⁡u|q{-\Delta_{m}u=u^{p}\lvert\nabla u\rvert^{q}} in ℝN{\mathbb{R}^{N}}, q≥m{q\geq m} and p>0{p>0}, 1<m<N{1<m<N}. Our main result is a Liouville-type property, namely, all the positive C1{C^{1}} solutions in ℝN{\mathbb{R}^{N}} are constant. We also give their asymptotic behaviour; all the solutions in an exterior domain ℝN∖Br0{\mathbb{R}^{N}\setminus B_{r_{0}}} are bounded. The solutions in Br0∖{0}{B_{r_{0}}\setminus\{0\}} can be extended as continuous functions in Br0{B_{r_{0}}}. The solutions in ℝN∖{0}{\mathbb{R}^{N}\setminus\{0\}} has a finite limit l≥0{l\geq 0} as |x|→∞{\lvert x\rvert\to\infty}. Our main argument is a Bernstein estimate of the gradient of a power of the solution, combined with a precise Osserman-type estimate for the equation satisfied by the gradient.

Abstract licence: CC BY

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