A problem involving the $p$-Laplacian operator

TL;DR

This paper establishes the existence of solutions for a class of nonlinear PDEs involving the p-Laplacian operator using variational methods, covering both sub-linear and super-linear cases with specific conditions.

Contribution

It provides a variational framework to guarantee solutions for the resonant Lane-Emden problem with the p-Laplacian, extending results to cases with additional forcing terms.

Findings

Existence of solutions under certain conditions for q in (1,p) and (p,p*)

Application of variational techniques to nonlinear p-Laplacian problems

Extension to cases with external forcing term f in L^{p'}()

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 $-\Delta_p u=\lambda |u|^{q-2}u+f$, $u|_{\partial\Omega}=0$, $f\in L^{p'}(\Omega)$ ($p'$ 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.