Non-existence of nodal solution for \(m\)-Laplace equation involving critical Sobolev exponents (Q1198388)

Let \(B_ R\) be a ball in \(\mathbb{R}^ n\) with radius \(R\). The authors consider the Dirichlet problem \[ -\Delta_ mu=| u|^{p- 1}u+| u|^{q-1}u\quad\text{ in } B_ R,\quad u=0\quad\text{ on } \partial B_ R, \tag{*} \] where \(\Delta_ mu:=\text{div}(|\nabla u|^{m-2}\nabla u)\) is the \(m\)-Laplacian, \(1<m<n\), \(p={nm-n+m\over n-m}\) is the critical Sobolev exponent, \(1\leq q\leq p\). If \(m-1\leq q\leq p-1\) the authors show, that there is a \(R_ 0>0\) such that for all \(0<R<R_ 0\) the Dirichlet problem \((*)\) admits no radial solution, which changes sign. In their proof they use a Pohozaev-type inequality and some facts from weighted eigenvalue inequalities for the \(m\)-Laplacian.