A sharp solvability condition in higher dimensions for some Brezis-Nirenberg type equation (Q1606054)

This paper is devoted to subcritical perturbations of quasilinear elliptic equations with Sobolev critical growth \[ \begin{cases} -\text{div} \bigl(|Du|^{p-2} Du\bigr)= u^{p^*-1}+g(u) \quad &\text{in } \Omega \\ u\geq 0\quad &\text{in }\Omega\\ u=0\quad &\text{on }\partial \Omega, \end{cases} \] where \(\Omega\subset\mathbb{R}^N\) is a bounded domain with smooth boundary, \(p^*={pN\over N-p}\), \(1<p<N\) and \(g\) has subcritical growth. The authors are mainly interested in a solution of mountain pass type. The main difficulty is due to the fact, that the usual test functions do not provide useful estimates for the mountain pass level. Here, the authors propose more suitable test functions which in fact give the proper bound.