On the Fučik spectrum for the $ p,q $-Laplacian

Here, we study the Fučik spectrum for $ p, q $-Laplacian$ -\Delta_p u-\Delta_q u = \alpha \left((u^+)^{p-1}+(u^+)^{q-1}\right)-\beta \left((u^-)^{p-1}+(u^-)^{q-1}\right) $in a bounded domain \Omega in \mathbb{R}^N ($ N\geq1 $), with Dirichlet boundary condition and 1<q<p<\infty $.