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 $\lambda_{ q}$.
Categories: math.AP