Bubble towers for supercritical semilinear elliptic equations

We construct positive solutions of the semilinear elliptic problem Δu+λu+up=0 with Dirichet boundary conditions, in a bounded smooth domain Ω⊂RN(N⩾4), when the exponent p is supercritical and close enough to N+2N-2 and the parameter λ∈R is small enough. As p→N+2N-2, the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves Green's function. Our result extends the result of Del Pino et al. (J. Differential Equations 193(2) (2003) 280) when Ω is a ball and the solutions are radially symmetric.