On the influence of second order uniformly elliptic operators in nonlinear problems

The purpose of the present paper is to discuss the role of second order elliptic operators of the type L = ∑ i , j = 1 n D i ( a i j( x ) D j )on the existence of a positive solution for the problem involving critical exponent− L u=u n + 2 n − 2+ λ uinΩ ,u = 0on ∂ Ω ,where Ω is a smooth bounded domain in R n , n ≥ 3 , and λ is a real parameter. In particular, we show that if the function det ( A ( x ) )has an interior global minimum point x 0 such that A ( x ) is comparable to A ( x 0 ) + | x − x 0 | γ I n , where A ( x ) = ( a i j( x ) ) and I n is the identity matrix of order n , then the range of values of λ for which the problem above has a positive solution can change drastically from γ ≤ 2 to γ > 2 .