Nodal solutions of semilinear elliptic equations with critical exponent

Let S1 C ~N be a bounded open set with smooth boundary and p = 2N/(N -2) be the critical Sobolev exponent.In this note we extend the results of [10] and [21] concerning nodal solutions (i.e., a solution which changes sign) for the Dirichlet problem: -LI.u = luiP-2 u + ..\u on S1 and u = 0 on 8S1, when N 2: 6 and ,.\ E (0, ..\1) with ..\1 the first eigenvalue of -LI. in HJ(Sl).Similarly, for the problem -LI.u = luiP-2 u + ..\jujq-2 u on S1 and u = 0 on 8S1 we obtain a nodal solution when,.\>0, (N + 2)/(N-2) < q < 2N/(N-2) for N = 3, 4, 5 and 2 < q < 2N/(N-2) for N 2: 6..>... depending on the domain 0 (see [6]).We refer to [4] and [18] for a detailed bibliography related to various interesting aspects of this problem.In this note, we will be concerned with solutions of ( *) which change sign in 0.Following the notation introduced in [2], we shall refer to these solutions as nodal solutions.The existence of a pair of nodal solutions for ( *) has been obtained in [10] and [21] for N ;::: 6 and .A E (0, .AI).Here we give a different proof of these results together with a mild extension.As in [10] and [21], we use variational methods.However, our proof relies more on the specific choice of the P.S. sequence then on the appropriate minimax principle.\Ve hope that our point of view will shed some new light on the multiplicity question for problem ( * ).