Asymptotic behavior of the solutions of some differential equations (Q2879995)

The work is devoted to the investigation of differential equations with partitional derivatives of the special type. It is obtained the analogue of the Liouville theorem for the solutions of it. In the open domain \({\mathcal O}\) of the space \({\mathbb C}^n\) it is considered the equation \(P(\mathfrak{L})f=0,\) where \(P\) denotes some non-zero given a polynomial, \(\mathfrak{L}\) is the so-called distorted Laplacian and \(f\) is unknown function. The main results of the paper are the following. For a domain \({\mathcal O}={\mathbb C}^n,\) theorem 1 states that the space of all locally integrable distributions \(f\) satisfying the condition \(P(\mathfrak{L})f=0\) and having the conditions on the growth of the special type consists only from the unit function \(f\equiv 0.\) Theorem 2 generalizes the statement of the Theorem 1 for the case, when a domain \({\mathcal O}\) is exterior of some compactum. Finally, theorem 3 states that the conditions of the theorems 1 and 2 are precise in some sense. The obtained results are applicable to various questions of mathematical analysis and differential equations.