Multiplicity of positive solutions for a p-q-laplacian type equation with critical nonlinearities

We study the effect of the coefficient f (x) of the critical nonlinearity on the number of positive solutions for a p - q -Laplacian equation. Under suitable assumptions for f (x) and g (x), we should prove that for sufficiently small > 0, there exist at least k positive solutions of the following p - q -Laplacian equation, - p u - q u = f x u | p - 2 u + g x u | r - 2 u in Ω, u = 0 on ∂ Ω, where Ω ⊂ R N is a bounded smooth domain, N > p, 1 < q < N (p - 1) / (N - 1) < p ≤ max { p, p- q / (p - 1) } < r < p ,p = N p / (N - p) is the critical Sobolev exponent, and s u = d i v (| u | s - 2 u is the s -Laplacian of u.