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On the first eigenvalue of the normalized \(p\)-Laplacian. (English) Zbl 1435.35184

The authors consider the eigenvalue problem \[ \begin{cases} -\Delta^N_p u = \lambda_pu \quad &\mbox{in}\;\; \Omega, \cr u = 0 \quad &\text{on}\;\; \partial\Omega, \end{cases}\tag{1} \] where \(\Omega\) is an open bounded subset of \(\mathbb{R}^n\), \(\Delta^N_p\) denotes the normalized or game-theoretic \(p\)-Laplacian, defined for any \(p \in (1,+\infty )\) by \[ \Delta^N_p u := \frac{1}{p}|\nabla u|^{2-p}\operatorname{div}(|\nabla u|^{p-2}\nabla u) \] The first eigenvalue of \(\Delta^N_p\) in \(\Omega\) is defined as \[ \overline{\lambda}_p(\Omega)\! :=\! \sup\{\lambda_p \!\in\! \mathbb{R} \,:\, \exists u\!>\!0 \;\text{such that}\; \Delta^N_p u+\lambda_p u\!\le\! 0 \;\text{in the viscosity sense}\}. \] The main result of this article is the following theorem.
Theorem 1. Let \(\Omega\) be an open bounded domain in \(\mathbb{R}^n\), with \(\partial\Omega\) is of class \(C^{2,\alpha }\) and connected. If \(u\) and \(v\) are two positive eigenfunctions associated with \(\overline{\lambda}_p(\Omega)\), then \(u\) and \(v\) are proportional, that is, there exists \(t \in \mathbb{R}_+\) such that \(u = tv\) in \(\Omega\).
In order to obtain Theorem 1 the authors follow the approach used by S. Sakaguchi [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 14, No. 3, 403–421 (1987; Zbl 0665.35025)]. As fundamental preliminary tools a Hopf-type lemma and the strict positivity of the eigenvalue are exloited.
The authors also find the following lower estimate for \(\overline{\lambda}_p(\Omega)\) in terms of the Lebesgue measure of \(\Omega\).
Theorem 3. For every open bounded domain \(\Omega\) in \(\mathbb{R}^n\) we have the lower bound \[ \overline{\lambda}_p(\Omega)\ge K_{n,p}|\Omega|^{-2/n}\,, \] with \[ K_{n,p} := \frac{(n[(p - 1) \wedge 1])^2}{p(p - 1)} 4^{-1+1/n} \pi^{1+1/n} \Gamma \biggl(\frac{n + 1}{2}\biggr)^{-2/n}. \]
The proof of Theorem 3 is obtained by the Alexandrov-Bakelman-Pucci method, as addressed by X. Cabré [Chin. Ann. Math., Ser. B 38, No. 1, 201–214 (2017; Zbl 1366.49053)].

MSC:

35J92 Quasilinear elliptic equations with \(p\)-Laplacian
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
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References:

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