Growth of meromorphic solutions and delay partial differential equations

Abstract It is shown that, out of a large class of delay partial differential equations, there is only a narrow class of equations, containing the Kac–Van Moerbeke equation, that can admit meromorphic solutions w satisfying the growth condition $$\mathop {\lim \sup}\limits_{r \to \infty} {{\log T({r,w})} \over r} = 0,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:munder> <mml:mrow> <mml:mo>lim</mml:mo> <mml:mo>sup</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:munder> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mi>log</mml:mi> <mml:mi>T</mml:mi> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mi>w</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>r</mml:mi> </mml:mfrac> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> </mml:math> where T ( r, w ) is the Nevanlinna characteristic of w . Condition (1) is, in the sense of Vojta’s dictionary, a direct analogue of the zero algebraic entropy criterion introduced by Hietarinta and Viallet.