Multiple Solutions for a Henon-Like Equation on the Annulus

For the equation (-\Delta u = | |x|-2 |^\alpha u^{p-1}), (1 < |x| < 3), we prove the existence of two solutions for (\alpha) large, and of two additional solutions when (p) is close to the critical Sobolev exponent (2^*=2N/(N-2)). A symmetry--breaking phenomenon appears, showing that the least--energy solutions cannot be radial functions.