Bubble towers for supercritical semilinear elliptic equations

Details Bubble towers for supercritical semilinear elliptic equations Bubble towers for supercritical semilinear elliptic equations

2004-09-09

arxiv.org/abs/math/0409153v1

Mathematics

Analysis of PDEs

Scientific paper

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 $\Omega \subset \R^N$ $(N\geq 4)$, when the exponent $p$ is supercritical and close enough to $\frac{N+2}{N-2}$ and the parameter $\lambda\in\R$ 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.

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