Abstract

Let B denote the unit ball in R N, N = 3. We consider the classical Brezis-Nirenberg problem(equation) where ? is a constant. It is proven in [3] that this problem has a classical solution if and only if ? < ? < ? 1 where A = 0 if N > 4, ? = ? 1/4 if N = 3. This solution is found to be unique in [17]. We prove that there is a number ? * and a continuous function a(?) = 0 decreasing in (?, ? *], increasing in [? *, ? 1) such that for each ? in this range and each µ ? (a(?),8) there exist a. µ-periodic function w µ(t) and two distinct radial solutions u µj-, j = 1, 2, singular at the origin, with u µj(x) ~ |x| -N-2/2w µ(log|x|) as x ? 0. They approach respectively zero and the classical solution as µ ? +8. At ? = ? * there is in addition to those above a solution ~ C N|x| -N-2/2. This clarifies a previous result by Benguria, Dolbeault and Esteban in [2], where a existence of a continuum of singular solutions for each ? ? (?, ? 1) was found.