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 $-{\Delta }_{p}u=\lambda |u{|}^{q-2}u$, $u{|}_{\partial \Omega }=0$ if and only if a solution to $-{\Delta }_{p}u=\lambda |u{|}^{q-2}u+f$, $u{|}_{\partial \Omega }=0$, $f\in L^{p^{\prime }}(\Omega )$ ($p^{\prime }$ 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.