SYMMETRY BREAKING FOR GROUND-STATE SOLUTIONS OF HENON SYSTEMS IN A BALL

We consider in this paper the problem {-Delta u = vertical bar x vertical bar(alpha) v(P), x is an element of Omega, -Delta u = vertical bar x vertical bar(beta) u(qs), x is an element of Omega, u > 0, v > 0, x is an element of Omega, u = v = 0, x is an element of partial derivative Omega, where Omega is the unit ball in R-N centred at the origin, 0 <= alpha < pN, beta > 0, N >= 3. Suppose q(epsilon) -> q as epsilon -> 0(+) and q epsilon, q satisfy, respectively, N/p + 1 + N/q(epsilon) + 1 > N-2, N/P + 1 + N/q + 1 = N - 2; we investigate the asymptotic estimates of the ground-state solutions (u(epsilon), v(epsilon)) of ( 1) as beta -> +infinity with p, q(epsilon) fixed. We also show the symmetry-breaking phenomenon with alpha, beta fixed and q(epsilon) -> q as epsilon -> 0(+). In addition, the ground-state solution is non-radial provided that epsilon > 0 is small or beta is large enough.