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Orthogonal polynomial solutions of an admissible potentially self-adjointlinear second-order partial $q$-difference equation of the hypergeometric typein two variables on $q$-linear lattices are analyzed. A $q$-Pearson's systemfor the orthogonality weight function, as well as for the differencederivatives of the solutions are presented, giving rise to a solution of the$q$-difference equation under study in terms of a Rodrigues-type formula. Themonic orthogonal polynomial solutions are treated in detail, giving explicitformulae for the matrices in the corresponding recurrence relations theysatisfy. Lewanowicz and Wo\'zny [S. Lewanowicz, P. Wo\'zny, J. Comput. Appl.Math. 233 (2010) 1554--1561] have recently introduced a (non-monic) bivariateextension of big $q$-Jacobi polynomials together with a partial $q$-differenceequation of the hypergeometric type that governs them. This equation isanalyzed in the last section: we provide two more orthogonal polynomialsolutions, namely, a second non-monic solution from the Rodrigues'representation, and the monic solution both from the recurrence relation thatgovern them and also explicitly given in terms of generalized bivariate basichypergeometric series. Limit relations as $q \uparrow 1$ for the partial$q$-difference equation and for the all three $q$-orthogonal polynomialsolutions are also presented.

Linear partial $q$-difference equations on $q$-linear lattices and their bivariate $q$-orthogonal polynomial solutions