We consider an elliptic system involving critical growth conditions. We develop a technique of variational methods for elliptic systems. Using the well-known results of maximum principle for systems developed by Fleckinger et al. (1995), we can find positive solutions. Also, we generalize the systems results obtained (for the scalar case) by Brézis and Nirenberg (1983). Also, we give applications to biharmonic equations.
Mario Zuluaga. "A note on the variational structure of an elliptic system involving critical Sobolev exponent." J. Appl. Math. 2003 (5) 227 - 241, 30 April 2003. https://doi.org/10.1155/S1110757X03208032