This paper is devoted to exploring the solutions of the product type nonlinear partial differential equations (PDEs) with three complex variables. By making use of Nevanlinna theory with several complex variables and Hadamard factorization theory of meromorphic functions, and combining with the properties of the full rank determinants and algebraic cofactor, we prove that equation∏i=13(ai1uz1+ai2uz2+ai3uz3)=1 in C3 has no any finite order transcendental entire solution under the condition that the rank of the following matrixA=(a11a12a13a21a22a23a31a32a33) is full rank, i.e. R(A)=3, where aij(i,j=1,2,3) are constants in C. Our results are some improvements and generalizations of the previous results given by Saleeby, Li, Lü and Xu. Meantime, we list some examples to explain that the condition in our theorem can not be removed.