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

Abstract

Let $\Omega$ be a bounded and smooth domain of $\mathbb{R}^{N}$, $N\geq2$, 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 $\partial\Omega,$ where $p>1$, $1\leq q<p^{\star}$ and $p^{\star}$ is the critical exponent of the Sobolev embedding $W_{0}^{1,p}(\Omega)\hookrightarrow L^{q}(\Omega)$. Two sequences, $\left( \lambda_{n}\right)_{n\in N}% \subset(0,\infty)$ 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\} $. It is shown that $\left( \lambda_{n}\right) _{n\in N}$ converges monotonically to an eigenvalue $\lambda\geq\lambda_{q}$, 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 $W_{0}^{1,p}(\Omega)$ to an eigenfunction $w$ corresponding to $\lambda$. The advantage of this method is that it can be used to find eigenvalues other than $\lambda_{q}$.