Existence of positive solutions to nonlinear elliptic equations involving critical Sobolev exponents

Abstract

In this paper we extend the results of Brezis and Nirenberg in [4] to the problem $$\left\{ \begin{gathered} Lu = - D_i (a_{ij} (x)D_j u) = b(x)u^p + f(x,u) in\Omega , \hfill \\ p = (n + 2)/(n - 2) \hfill \\ u > 0 in\Omega , u = 0 \partial \Omega , \hfill \\ \end{gathered} \right.$$ whereL is a uniformly elliptic operator,b(x)≥0,f(x,u) is a lower order perturbation ofup at infinity. The existence of solutions to (A) is strongly dependent on the behaviour ofaij (x), b(x) andf(x, u). For example, for any bounded smooth domain Ω, we have\(a_{ij} \left( x \right) \in C\left( {\bar \Omega } \right)\) such thatLu=up possesses a positive solution inH01(Ω).