Bobkov, Vladimir; Drábek, Pavel; Ilyasov, Yavdat Estimates on the spectral interval of validity of the anti-maximum principle. (English) Zbl 1439.35105 J. Differ. Equations 269, No. 4, 2956-2976 (2020). Summary: The anti-maximum principle for the homogeneous Dirichlet problem to \(- \Delta_p u = \lambda | u |^{p - 2} u + f(x)\) with positive \(f \in L^{\infty}(\Omega)\) states the existence of a critical value \(\lambda_f > \lambda_1\) such that any solution of this problem with \(\lambda \in( \lambda_1, \lambda_f)\) is strictly negative. In this paper, we give a variational upper bound for \(\lambda_f\) and study its properties. As an important supplementary result, we investigate the branch of ground state solutions of the considered boundary value problem in \(( \lambda_1, \lambda_2)\). Cited in 1 Document MSC: 35B50 Maximum principles in context of PDEs 35B09 Positive solutions to PDEs 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35B38 Critical points of functionals in context of PDEs (e.g., energy functionals) 35J25 Boundary value problems for second-order elliptic equations 35J92 Quasilinear elliptic equations with \(p\)-Laplacian Keywords:\(p\)-Laplacian; ground state; nodal solutions × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Allegretto, W.; Huang, Y. X., A Picone’s identity for the p-Laplacian and applications, Nonlinear Anal., Theory Methods Appl., 32, 7, 819-830 (1998) · Zbl 0930.35053 [2] Anane, A., Simplicité et isolation de la premiere valeur propre du p-laplacien avec poids, C. R. Acad. 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