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.

中文翻译：

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具有超临界源梯度项的拟线性椭圆型方程的Liouville结果和渐近解

我们考虑ℝN{\ mathbb中的椭圆拟线性方程-Δm⁢u=up⁢|∇⁡u| q {-\ Delta_ {m} u = u ^ {p} \ lvert \ nabla u \ rvert ^ {q}} {R} ^ {N}}，q≥m{q \ geq m}和p> 0 {p> 0}，1 <m <N {1 <m <N}。我们的主要结果是一个Liouville型性质，即ℝN{\ mathbb {R} ^ {N}}中所有正C1 {C ^ {1}}解都是恒定的。我们还给出了它们的渐近行为。外部域ℝN∖Br0 {\ mathbb {R} ^ {N} \ setminus B_ {r_ {0}}}中的所有解都是有界的。Br0∖{0} {B_ {r_ {0}} \ setminus \ {0 \}}中的解可以扩展为Br0 {B_ {r_ {0}}}中的连续函数。ℝN∖{0} {\ mathbb {R} ^ {N} \ setminus \ {0 \}}中的解具有| x |→∞{\ lvert x \ rvert \ to \ infty}。我们的主要论据是对解决方案功效梯度的伯恩斯坦估计，并结合由梯度满足的方程的精确Osserman型估计。