SYMMETRY AND STABILITY OF NON-NEGATIVE SOLUTIONS TO DEGENERATE ELLIPTIC EQUATIONS IN A BALL

We consider non-negative distributional solutions u is an element of C-1((B-R) over bar) to the equation - div[g(vertical bar del u vertical bar)vertical bar del u vertical bar(-1)del u] = f(vertical bar x vertical bar, u) in a ball B-R, with u = 0 on partial derivative B-R, where f is continuous and non-increasing in the first variable and g is an element of C-1(0, +infinity) boolean AND C[0, +infinity), with g(0) = 0 and g'(t) > 0 fort > 0. According to a result of the first author, the solutions satisfy a certain 'local' type of symmetry. Using this, we first prove that the solutions are radially symmetric provided that f satisfies appropriate growth conditions near its zeros. In a second part we study the autonomous case, f = f (u). The solutions of the equation are critical points for an associated variation problem. We show under rather mild conditions that global and local minimizers of the variational problem are radial.