Global solutions of homogeneous linear partial differential equations of the second order (Q846236)

The authors study meromorphic solutions of homogeneous partial differential equations of the second order in two complex variables. They consider the equation \[ a_0\partial^2 u/\partial t^2 + 2a_1\partial^2 u/\partial t\partial z+ a_2\partial^2 u/\partial z^2+a_3\partial u/\partial t+a_4\partial u/\partial z+ a_5 u=0,\tag{A} \] where \(a_0, \dots, a_5\) are holomorphic functions in a region \(\Sigma \subseteq \mathbb{C}^2\). They first consider the special case where the equation is \(t^2\partial^2 u/\partial t^2-\partial^2 u/\partial z^2+t\partial u/\partial t-z\partial u/\partial z+t^2 u=0\) and obtain a condition for which the above equation has an entire solution. They also give results for entire solutions of the above equation from a view point of Nevanlinna theory. In general case, they give some unicity theorems for the meromorphic solution of the equation (A) and a theorem on deficiency of the meromorphic solution.