The sign of Lagrange multiplier for some minimization problem (Q2276745)

We consider the minimization problem: \[ (P)\quad \inf_{u\in H^ 1_ 0(\Omega),\| u+\phi \|_{L^{2^*}}=1}\int_{\Omega}| \nabla u|^ 2=J, \] where \(\Omega\) is a smooth bounded domain in \({\mathbb{R}}^ N\) (N\(\geq 3)\), \(2^*=(2N/(N-2))\) is the critical Sobolev exponent. We make the hypothesis that \(\phi\not\equiv 0\) and we show that if \(0<\| \phi \|_{L^{2^*}}<1\), the Lagrange multiplier associated to (P) is positive, when \(\| \phi \|_{L^{2^*}}>1\), it is negative and equal to zero when \(\| \phi \|_{L^{2^*}}=1\).