A note on a second order PDE with critical nonlinearity

In this work, we are interested in a nonlinear PDE of the form: −∆u = K(x)u n+2 n−2 , u > 0 on Ω and u = 0 on ∂Ω, where n ≥ 3 and Ω is a regular bounded domain of Rn . Following the results of [K. Sharaf, Appl. Anal. 96(2017), No. 9, 1466– 1482] and [K. Sharaf, On an elliptic boundary value problem with critical exponent, Turk. J. Math., to appear], we provide a full description of the loss of compactness of the problem and we establish a general index account formula of existence result, when the flatness order of the function K at any of its critical points lies in (1, ∞)