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 $-\Delta_p u=\lambda |u|^{q-2}u$, $u|_{\partial\Omega}=0$ 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 $q\in (1,p)\bigcup (p,p^{*})$ under a certain condition for both the cases, i.e., $1<q<p<p^{*}$ and $1< p < q < p^{*}$ - the sub-linear and the super-linear cases. 
Ratan K. Giri, D. Choudhuri
math.AP