A nonexistence result for sign-changing solutions of the Brezis-Nirenberg problem in low dimensions (Q2344708)

The Brezis-Nirenberg problem \[ \begin{cases} -\Delta u = \lambda u + |u|^{2^*-2} u & \text{in} \;\Omega, \\ u=0 & \text{on}\, \partial \Omega, \end{cases} \] is considered, where \(\Omega\) is a smooth bounded domain in \({\mathbb R^N}\), \(N\geq 3\), \(2^*=2N/(N-2)\) is the critical Sobolev exponent and \(\lambda>0\) is a positive parameter. It is shown that if \(N=4\), \(5\), \(6\) and \(\lambda\) is close to \(0\), there are no sign-changing solutions of the form \(u_\lambda = P U_{\delta_1,\xi} - P U_{\delta_2,\xi} + w_\lambda\), where \(P U_{\delta_i,\xi}\), \(i=1\), \(2\) is the projection on \(H_0^1(\Omega)\) of the regular positive solution of the critical problem in \({\mathbb R}^N\), centered at a point \(\xi \in \Omega\) and \(w_\lambda\) is a remainder term. Some additional results on norm estimates of \(w_\lambda\) and about the concentrations speeds of tower of bubbles in higher dimensions are also presented.