Degenerate elliptic inequalities with critical growth

This article is motivated by the fact that very little is known about variational inequalities of general principal differential operators with critical growth.The concentration compactness principle of P.L. Lions [P.L. Lions, The concentration compactness principle in the calculus of variation. The limit case I, Rev. Mat. Iberoamericana 1 (1) (1985) 145–201; P.L. Lions, The concentration compactness principle in the calculus of variation. The limit case II, Rev. Mat. Iberoamericana 1 (2) (1985) 45–121] is a widely applied technique in the analysis of Palais–Smale sequences. For critical growth problems involving principal differential operators Laplacian or p-Laplacian, much has been accomplished in recent years, whereas very little has been done for problems involving more general main differential operators since a nonlinearity is observed between the corresponding functional I(u) and measure μ introduced in the concentration compactness method. In this paper, we investigate a Leray–Lions type operator and behaviors of its (P.S.)c sequence.