Analysis reveals finite order entire solutions for nonlinear partial differential equations, suggesting matrix transformations can provide clarity on existing questions.
In this paper, we describe finite order entire solutions of nonlinear partial differential equation {equation*} ∏ ᵢ₌₁^n (ai 1 uz_1+ai 2 uz_2+⋯ +ai n uz_n )=1 {equation*} with the eikonal equation as a prototype. By the elementary row transformation of a matrix, we offer an affirmative answer to a question of Xu-Liu-Xuan in Xu et al. ( J. Math. Anal. Appl. 543 (2025), ID 128885, 21 p.).
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Lü et al. (2025) studied this question.
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