A problem involving the $p$-Laplacian operator

Abstract

Using a variational technique we guarantee the existence of a solution to the \emph{resonant Lane-Emden} problem $-Δ_p u=λ|u|^{q-2}u$, $u|_{\partialΩ}=0$ if and only if a solution to $-Δ_p u=λ|u|^{q-2}u+f$, $u|_{\partialΩ}=0$, $f\in L^{p'}(Ω)$ ($p'$ 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.