García-Melián, J.; Sabina de Lis, J. Maximum and comparison principles for operators involving the \(p\)-Laplacian. (English) Zbl 0897.35015 J. Math. Anal. Appl. 218, No. 1, 49-65 (1998). This paper is concerned with positivity properties of the operator \[ {\mathfrak L}_{p,a}: =-\Delta_p u+a (\cdot)| u|^{p-2} u,\;p>1, \] under homogeneous Dirichlet conditions in the bounded smooth domain \(\Omega\subset \mathbb{R}^n\). Here \(-\Delta_pu =-\text{div} (|\nabla u|^{p-2} \nabla u)\) is the \(p\)-Laplace operator and \(a(\cdot)\in L^\infty (\Omega)\). While positivity and even comparison results are well known if \(a\geq 0\) [see P. Tolksdorff, J. Differ. Equations 51, 126-150 (1984; Zbl 0522.35018)], the authors are mainly interested in the case \(a\ngeq 0\). Among a lot of other results they show that the validity of a strong positivity result \(0\not \equiv {\mathfrak L}_{p,a} u\geq 0\) in \(\Omega\), \(u|\partial \Omega \geq 0\Rightarrow u>0\) in \(\Omega\) is equivalent to the positivity of the first eigenvalue \[ \lambda_{1,p}(a) {\overset{def}=} \inf\biggl\{ \int_\Omega\bigl( | \nabla u|^p+ a| u|^p\bigr) dx:u\in W_0^{1,p} (\Omega),\;\int_\Omega| u|^pdx =1\biggr\}>0. \] Reviewer: H.-Ch.Grunau (Bayreuth) Cited in 53 Documents MSC: 35B50 Maximum principles in context of PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations 35J70 Degenerate elliptic equations Keywords:strong positivity; first eigenvalue Citations:Zbl 0522.35018 PDFBibTeX XMLCite \textit{J. García-Melián} and \textit{J. Sabina de Lis}, J. Math. Anal. 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