An eigenvalue problem for the Dirichlet ( <i>p</i> , <i>q</i> )-Laplacian (Q6584618)

This article is concerned with the eigenvalue problem \(-\Delta_p u-\Delta_q u=\lambda(u^\tau-u^{\mu-1})\) in \(\Omega\), subject to homogeneous Dirichlet boundary condition. In the above, \(\Omega\subset\mathbb{R}^N\) is a \(C^2\) bounded domain, \(1<\mu<\tau<q<p\), \(\lambda>0\). The main result establishes the existence of \(\lambda_*>0\) such that:\N\begin{itemize}\N\item for all \(0<\lambda<\lambda_*\) the problem has no positive solutions;\N\item for all \(\lambda\geq \lambda_*\) there exist two positive and ordered solutions.\N\end{itemize}\NThe approach is variational and uses truncation techniques.