SINGULAR SOLUTIONS OF THE BREZIS-NIRENBERG PROBLEM IN A BALL

Let B denote the unit ball in R-N, N >= 3. We consider the classical Brezis-Nirenberg problem {Delta u + lambda u + u(N+2/N-2) = 0 in B u > 0 in B u = 0 on partial derivative B where lambda is a constant. It is proven in [3] that this problem has a classical solution if and only if (lambda) under bar < lambda < lambda(1) where (lambda) under bar = 0 if N >= 4, (lambda) under bar = lambda(1)/4 if N = 3. This solution is found to be unique in [17]. We prove that there is a number lambda(*) and a continuous function a(lambda) >= 0 decreasing in ((lambda) under bar, lambda(*)], increasing in [lambda(*),lambda(1)) such that for each lambda in this range and each mu is an element of (a(lambda), infinity) there exist a mu-periodic function w(mu)(t) and two distinct radial solutions u(mu j), j = 1, 2, singular at the origin, with u(mu j)(x) similar to |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 lambda is an element of ((lambda) under bar, lambda(1)) was found.