Multiple solutions to anisotropic critical and supercritical problems in symmetric domains

Abstract

We consider the problem

$$\displaystyle{-\mbox{ div}(a(x)\nabla u) + b(x)u = c(x)\vert u\vert ^{p-2}u\text{ in }\varOmega,\ \ u = 0\text{ on }\partial \varOmega,}$$

where Ω is a bounded smooth domain Ω in $\mathbb{R}^{N}$ and $p = \frac{2(N-k)} {N-k-2},$ 0 ≤ k ≤ N − 3, is the (k + 1)-st critical exponent. We establish existence of a prescribed number of solutions to this problem in domains of revolution, under some symmetry assumptions. For p = 2^∗ we prove a global compactness result for the G-invariant Palais-Smale sequences of the variational functional associated with this problem, which relates the symmetries of the concentration points to those of the solution to the limit problem that concentrates at those points.