Sign-changing solutions of an elliptic system with critical exponent in dimension N=5

We study the following elliptic system with critical exponent: {- u =.1u + mu 1| u| 2 *- 2u + ss| u| 2 * 2 2u| v| 2 * 2, x. , - v =.2v + mu 2| v| 2 *- 2v + ss| v| 2 * 2 - 2v| u| 2 * 2, x. , u = v = 0, x.. , where is the first eigenvalue of in H01(). In [10], Chen, Lin and Zou established a sign-changing solution of the above system in the case N 6 for < 0 and (1), (2) (0, (1)()). We show that in dimension N = 5, for (1) and (2) slightly smaller than (1)(), the above system has a sign-changing solution in the following sense: one component changes sign and has exactly two nodal domains, while the other one is positive.