On positive solutions of indefinite elliptic equations (Q2756224)

The goal of this note is to determine under which conditions on the positive number \(\lambda\) one can or one cannot find a positive solution of NEWLINE\[NEWLINE \begin{cases} -\Delta_p u=\lambda |u|^{p-2} u+f(x)|u|^{\gamma-2} u,\;u>0\text{ in }\Omega,\\ u=0\quad \text{on }\partial \Omega, \end{cases} NEWLINE\]NEWLINE where \(\Omega\) is bounded and smooth domain of \(\mathbb{R}^n\), let \(\Delta_p\) denote the \(p\)-Laplacian defined by: \(\Delta_p= \text{div}(|\nabla u |^{p-2} \nabla u)\) for \(p\in (1,\infty)\), \(1<p< \gamma\leq p^*\), with \(p^*= {pn\over n-p}\) if \(p<n\) and \(p^*= +\infty\) if \(p\geq n\).