Pergamon Nonlrnenr Anolyris, Theory. Methods 8s Apphcofmns, Vol. 26, No. 5, pp. 971-995. 1996 Copyright ‘D 1995 Elsemer Science Ltd Printed in Great Britain. All rights reserved 0362-546X/96 $15.W+ .OO 0362-546X(94)00301-7 CRITICAL BEHAVIOR OF SEMI-LINEAR ELLIPTIC EQUATIONS WITH SUB-CRITICAL EXPONENTS GERSHON WOLANSKY Department of Mathematics, Technion, Israel Institute of Technology, Haifa 32000, Israel (Received 6 April 1994; received for publication 19 October 1994) Key words and phrases: Boundary value problem, critical exponent, blow-up singularity, symmetry breaking. 1. INTRODUCTION Let Q be a bounded domain in P with a smooth boundary. Consider the boundary value problem of the type (4 @I AU + g(U,x) = 0 on fi U=A on X& J is an unknown constant (1.1) ’ au ,,anapl = -MT I where g(*, a) 2 0 on It?’ 0 0 and M > 0 is a prescribed constant. Equation (1.1) is a peculiar type of an elliptic, semi-linear boundary problem which involves the integral constraint (c). There are several motivations for the study of this type of problem, e.g. (i) if 0 c F?’and g(t) = 0 for t s 0, then (1.1) is a generalization of Grad’s formulation of a model for confined plasma in a magnetic bottle. Here M stands for the total magnetic flux (cf. [l] and the appendix in [2]); (ii) for a two dimensional domain and g(t, x) = K(x) e’, (1.2) the solutions of (1.1) yield Riemannian metrics which are conformally equivalent to the Euclidean metric on the two dimensional domain Q with a given Gaussian curvature K and a prescribed total curvature M/2 [3,4]; (iii) if g(t, x) = f(t - V(x)), (1.3) where V E C’(a) is given and f( *) is nondecreasing, then the solutions of (1.1) represent potentials of stationary distributions of self-gravitating clusters. Here M stands for the total mass of the cluster [5]. The existence of solutions for (1.1) in the caseg := g(t) = pt+ (t+ := the positive part of t) was first established by Berestycki and Brezis [6] and later by Temam [2]. In [7], Berestycki and Brezis extended the existence proof for more general functions g( *, t) which satisfy (1.4) 971