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 ${\mathbb{R}}^N$, $N\ge 2$, 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 $\partial \Omega ,$ where $p>1$, $1\le q<p^{\star }$ and $p^{\star }$ is the critical exponent of the Sobolev embedding $W_0^{1,p}(\Omega )\hookrightarrow L^q(\Omega )$. Two sequences, ${\left({\lambda }_n\right)}_{n\in N}$ 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 $\lambda \ge {\lambda }_q$, 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 $W_0^{1,p}(\Omega )$ to an eigenfunction $w$ corresponding to $\lambda$. The advantage of this method is that it can be used to find eigenvalues other than ${\lambda }_q$.

This paper will not be published in other source. It was considerably improved, giving rise to a new paper, by the same author, titled "Solving an abstract nonlinear eigenvalue problem by the inverse iteration method", arXiv:1708.00116

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