ON THE REGULARITY AND PARTIAL REGULARITY OF EXTREMAL SOLUTIONS OF A LANE EMDEN SYSTEM

In this paper, we consider the system - Delta u = lambda(v + 1)(p), -Delta v = gamma(u + 1)(theta) on a smooth bounded domain Omega in R-N with Dirichlet boundary condition u = v = 0 on partial derivative Omega. Here lambda, gamma are positive parameters and 1 < p <= theta. Let x0 be the largest root of the polynomial H(x) = x(4)-16p theta(p+1)(theta+1)/(p(0) - 1)(2)x(2) + 16p theta(p_1) (theta + 1) (p+theta+2)/(p(theta) - 1)(3)x -16p theta(p + 1)(2)(theta + 1)(2)/(p theta - 1)(4). We show that the extremal solutions associated to the above system are bounded provided N < 2 + 2x(0). This improves the previous work by Craig Cowan (2015). We also prove that if N >= 2 + 2x(0), then the singular set of any extremal solution has Hausdorff dimension less than or equal to N (2+2x(0)).