SOME REMARKS ON C(1) VERSUS H(1) MINIMIZE RS

In a recent paper, Brezis and Nirenberg [2] considered an elliptic Dir ichlet problem with at most critical growth as the Euler-Lagrange equa tion of a well-defined ''action functional'' in H0(1). They show that any local minimizer for this functional in the C1-topology must corres pond to a local minimizer in the H0(1)-topology. The aim of this Note is to show how, according to the nonlinearity, this might or might not be the case for elliptic problems with a well-defined variational str ucture in H0(1), but admitting supercritical growth.