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

Abstract : We construct solutions of the semilinear elliptic problem {Delta u + \u\(p-1)u+epsilon(1)(2) f = 0 in Omega u = epsilon(1)(2) g on partial derivative Omega in a bounded smooth domain Omega subset of R-N ( N >= 3), when the exponent p is supercritical and close enough to N+2/N-2. As p --> N+2/N-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. As applications, we will give some existence results, in particular, when Omega are symmetric domains perforated with the small hole and when f = 0 and g = 0.