Bubble tower solutions of slightly supercritical elliptic equations and application in symmetric domains

We construct solutions of the semilinear elliptic problem {Δu+ |u|^P-1u + ε1/2 f = 0 in Ω u =ε1/2g on ℓΩ in a bounded smooth domain Ω ⊂ ℝ^N (N ≥ 3), when the exponent p is supercritical and close enough to N+2/N-2. As p → N+2/N-1 the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves Green's function. As applications, we will give some existence results, in particular, when Ω are symmetric domains perforated with the small hole and when f = 0 and g = 0.