Bubble towers for supercritical semilinear elliptic equations

Abstract

We construct positive solutions of the semilinear elliptic problem $\Delta u+\lambda u+u^p=0$ with Dirichet boundary conditions, in a bounded smooth domain $\Omega \subset {\mathbb{R}}^N$ $(N\ge 4)$, when the exponent $p$ is supercritical and close enough to $\frac{N+2}{N-2}$ and the parameter $\lambda \in \mathbb{R}$ is small enough. As $p\to \frac{N+2}{N-2}$, the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves Green's function. Our result extends the result of Del Pino, Dolbeault and Musso \cite{DDM} when $\Omega$ is a ball and the solutions are radially symmetric.