We study an asymptotic behavior of solutions to elliptic equations of the second order in a two dimensional exterior domain. Under the assumption that the solution belongs to $$L^q$$ L q with $$q \in [2,\infty )$$ q ∈ [ 2 , ∞ ) , we prove a pointwise asymptotic estimate of the solution at the spatial infinity in terms of the behavior of the coefficients. As a corollary, we obtain the Liouville-type theorem in the case when the coefficients may grow at the spacial infinity. We also study a corresponding parabolic problem in the n -dimensional whole space and discuss the energy identity for solutions in $$L^q$$ L q . As a corollary we show also the Liouville-type theorem for both forward and ancient solutions.

中文翻译：

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在无界域中具有二阶无界系数的椭圆和抛物线方程解的渐近行为

我们研究了二维外部域中二阶椭圆方程解的渐近行为。在解属于 $$L^q$$L q 且 $$q \in [2,\infty )$$ q ∈ [ 2 , ∞ ) 的假设下，我们证明了在就系数的行为而言，空间无穷大。作为推论，我们在系数可能在空间无穷大增长的情况下获得了 Liouville 型定理。我们还研究了 n 维整个空间中相应的抛物线问题，并讨论了 $$L^q$$L q 中解的能量恒等式。作为推论，我们还展示了正解和古解的 Liouville 型定理。