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 lambda > 0, there exist at least k positive solutions of the following p-q-Laplacian equation, -Delta(p)u -Delta(q)u - f(x) vertical bar u vertical bar(p*-2)u+lambda g(x)vertical bar u vertical bar(r-2)u in Omega, u = 0 on partial derivative Omega, where Omega subset of 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* = Np/( N - p) is the critical Soboleve exponent, and Delta(s)u = div(vertical bar del u vertical bar(s-2)del u is the s-Laplacian of u.