An inverse iteration method for obtaining q-eigenpairs of the p-Laplacian in a general bounded domain

Let $\mathrm{\Omega }$ be a bounded and smooth domain of ${\mathbb{R}}^N$, $N\ge 2$, and consider the eigenvalue problem: $-\Delta_{p}u=\lambda\left| u\right| {L^{q}(\Omega)}^{p-q}\left| u\right| ^{q-2}u$in$\Omega,$ $u=0$ on $\mathrm{\partial }\mathrm{\Omega },$ where $p>1$, $1\le q<p^{\star }$ and $p^{\star }$ is the critical exponent of the Sobolev embedding $W{0}^{1,p}(\Omega)\hookrightarrow L^{q}(\Omega)$\mathrm{.}Twosequences,$\left( \lambda_{n}\right){n\in N}% \subset(0,\infty)$and$\left(w{n}\right) {n\in N}\subset W{0}^{1,p}(\Omega)$, are built by means of an inverse iteration scheme starting from an arbitrary function $u_{0}\in L^{q}(\Omega)\backslash\left{ 0\right} $\mathrm{.}Itisshownthat$\left( \lambda_{n}\right) _{n\in N}$ converges monotonically to an eigenvalue $\lambda \ge {\lambda }_q$, with ${\lambda }_q$ denoting the first eigenvalue. It is also proved that there exists a subsequence ${\left(w_{n_j}\right)}_{j\in \mathbb{N}}$ converging in $W_0^{1,p}(\mathrm{\Omega })$ to an eigenfunction $w$ corresponding to $\lambda$. The advantage of this method is that it can be used to find eigenvalues other than ${\lambda }_q$.