A problem involving the $p$-Laplacian operator

Using a variational technique we guarantee the existence of a solution to the \emph{resonant Lane-Emden} problem $-{\mathrm{\Delta }}_{p}u=\lambda \mathrm{\mid }u{\mathrm{\mid }}^{q-2}u$, $u{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=0$ if and only if a solution to $-\Delta_p u=\lambda |u|^{q-2}u+f$,$u|_{\partial\Omega}=0$,$f\in L^{p’}(\Omega)$($p’$ being the conjugate of $p$), exists for $q\in (1,p)\bigcup (p,p^{\ast })$ under a certain condition for both the cases, i.e., $1<q<p<p^{\ast }$ and $1<p<q<p^{\ast }$ - the sub-linear and the super-linear cases.