Nonexistence results of sign-changing solutions to a supercritical nonlinear problem (Q939215)

The following semi-linear elliptic problem with super critical nonlinearity \[ -\Delta u=| u| ^{p-1+\varepsilon} u \quad\text{in } \Omega, \qquad u=0 \quad\text{on }\partial\Omega, \leqno (P_\varepsilon) \] where \(\Omega\) is a smooth domain in \(\mathbb{R}^N, N\geq 3,\varepsilon>0\) and \(p=\frac{N+2}{N-2}\) is the critical Sobolev exponent, is considered in the paper. It is shown that for \(\varepsilon\) small, the problem \((P_\varepsilon)\) has no sign-changing solutions with low energy which blow up at two points. It is shown also that \((P_\varepsilon)\) has no bubble-tower sign-changing solutions.