Bounds for the positive eigenvalues of the \(p\)-Laplacian with decaying potential
The authors study the equation \[ (-r^{a}{u'}^{(p-1)})' + r^{a}q(r)u^{(p-1)} = \lambda r^{a}u^{(p-1)} \] for \(p>1\), \(a \geq 0\) and \(q(r) \rightarrow 0\) as \(r\rightarrow \infty\). Here, \(u^{(p-1)} = | u | ^{(p-2)} u\). Using a generalized Prüfer transformation, one obtains an explicit upper bound in terms of \(q\) for the positive values of the spectral parameter \(\lambda\) for which a solution \(u \in L^{p}(0,\infty; r^{a})\) of the equation exists. It is shown that this bound is optimal. These results generalize known spectral properties of the classical linear Laplacian [see e.g. \textit{F. V. Atkinson} and \textit{W. N. Everitt}, Proc. R. Soc. Edinb., Sect. A 80, 57--66 (1978; Zbl 0426.34015)].
scientific article; zbMATH DE number 5078962
scientific article