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

Abstract

Let $Ω$ be a bounded and smooth domain of $\mathbb{R}^{N}$, $N\geq2$, and consider the eigenvalue problem: $-Δ_{p}u=λ\left| u\right| _{L^{q}(Ω)}^{p-q}\left| u\right| ^{q-2}u$ in $Ω,$ $u=0$ on $\partialΩ,$ where $p>1$, $1\leq q<p^{\star}$ and $p^{\star}$ is the critical exponent of the Sobolev embedding $W_{0}^{1,p}(Ω)\hookrightarrow L^{q}(Ω)$. Two sequences, $\left( λ_{n}\right)_{n\in N}% \subset(0,\infty)$ and $\left(w_{n}\right) _{n\in N}\subset W_{0}^{1,p}(Ω)$, are built by means of an inverse iteration scheme starting from an arbitrary function $u_{0}\in L^{q}(Ω)\backslash\left\{ 0\right\} $. It is shown that $\left( λ_{n}\right) _{n\in N}$ converges monotonically to an eigenvalue $λ\geqλ_{q}$, with $λ_{q}$ denoting the first eigenvalue. It is also proved that there exists a subsequence $\left( w_{n_{j}}\right) _{j\in\mathbb{N}}$ converging in $W_{0}^{1,p}(Ω)$ to an eigenfunction $w$ corresponding to $λ$. The advantage of this method is that it can be used to find eigenvalues other than $λ_{q}$.