Abstract We consider the elliptic equation - Δ ⁢ u = u q ⁢ | ∇ ⁡ u | p {-\Delta u=u^{q}|\nabla u|^{p}} in ℝ n {\mathbb{R}^{n}} for any p > 2 {p>2} and q > 0 {q>0} . We prove a Liouville-type theorem, which asserts that any positive bounded solution is constant. The proof technique is based on monotonicity properties for the spherical averages of sub- and super-harmonic functions, combined with a gradient bound obtained by a local Bernstein argument. This solves, in the case of bounded solutions, a problem left open in [2], where the case 0 < p < 2 {0

中文翻译：

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梯度超二次增长椭圆方程的刘维尔型定理

摘要 我们考虑椭圆方程 - Δ ⁢ u = uq ⁢ | ∇ ⁡ u | p {-\Delta u=u^{q}|\nabla u|^{p}} in ℝ n {\mathbb{R}^{n}} 对于任何 p > 2 {p>2} 和 q > 0 {q>0}。我们证明了一个 Liouville 型定理，该定理断言任何正有界解都是常数。证明技术基于次谐波和超谐波函数的球面平均值的单调性，并结合由局部 Bernstein 参数获得的梯度界限。在有界解的情况下，这解决了 [2] 中未解决的问题，其中情况 0 < p < 2 {0