On solutions to systems involving a Paneitz–Branson type operator with critical growth

Abstract In this paper we investigate a class of nonlinear elliptic systems of fourth order on closed Riemannian manifolds, involving a Paneitz–Branson type operator and nonlinearities with critical Sobolev growth. Such systems combine the difficulties of higher order operators, strong coupling effects, and critical nonlinear behavior, and arise naturally in problems motivated by geometric analysis. By means of variational methods, we establish the existence of weak solutions under suitable structural assumptions on the coefficients of the operator and on the interaction between the lower order terms and the geometry of the manifold. The approach relies on a careful analysis of the associated energy functional and on comparisons with the classical Paneitz–Branson operator. In addition, by standard elliptic regularity theory, we show that the weak solutions obtained are in fact classical.