THE CRITICAL CURVE FOR PERIODIC DEGENERATE LANE-EMDEN TYPE SYSTEM

In this paper we are concerned with the critical curve for the existence of periodic solutions for the degenerate Lane-Emden type system. We give an optimal curve to divide the existence and nonexistence of nontrivial nonnegative periodic solutions. Exactly speaking, we propose a new curve n/p+n + m/q+m = N-2/N, which is proved to be the critical curve. Note that if m = n = 1, then it becomes the Sobolev hyperbola 1/p+1 + 1/q+1 = N-2/N, which is expected to be the critical curve for the well-known Lane-Emden conjecture. We also obtain a singular curve pq = mn, where the nontrivial nonnegative periodic solutions may exist or do not. Obviously, the usual methods for scalar parabolic equation cannot be used to the present coupled degenerate system directly. Moreover, the complex relationship between the exponents m, n, p and q also makes the proof of the theorems in this paper quite technical.