Abstract

In this article we present the analysis of critical exponents for a large class of extremal operators, in the case of radially symmetric solutions. More precisely, for such an operator M, we consider the nonlinear equation (*) M(D2u) + up = 0, u > 0 in ℝN and we prove the existence of a critical exponent p* that determines the range of p > 1 for which we have existence or non-existence of a positive radial solution to (*). In the case of maximal operators, we define two dimension-like numbers N∞ and N0, depending on M and N, that satisfy 0 < N∞ ≤N0. We prove that our critical exponent satisfies max {N∞/N∞-2, p0} ≤ p * ≤ p∞ where p0 = (N0 + 2)/(N0 - 2) and p∞ = (N ∞ + 2)/(N∞ - 2). In the non-trivial case, N∞ < N0 and both inequalities above are strict. Indiana University Mathematics Journal ©.