Brasco, Lorenzo; Franzina, Giovanni; Ruffini, Berardo Schrödinger operators with negative potentials and Lane-Emden densities. (English) Zbl 1388.35135 J. Funct. Anal. 274, No. 6, 1825-1863 (2018). The authors consider the smallest eigenvalue of the Schrödinger operator \(-\Delta+V\) subject to Dirichlet boundary conditions in an open set \(\Omega \subset \mathbb{R^N}\). The potential \(V\) is assumed to be negative. A condition is given that nevertheless the spectrum is positive in form of an explicit bound on \(V\). The bound is obtained with the aid of a Hardy-type inequality which uses a weight-function depending on the solution of the so-called Lane-Emden equation \(-\Delta u = u^{q-1}\) subject to Dirichlet boundary conditions, where \(1 \leq q <2\). Applications are given to special cases of \(\Omega\). Reviewer: Rolf Dieter Grigorieff (Berlin) Cited in 1 ReviewCited in 6 Documents MSC: 35P15 Estimates of eigenvalues in context of PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 49S05 Variational principles of physics Keywords:Poisson equation; negative potential; first eigenvalue; lower bound; Lane-Emden density; Hardy-type inequality; Sobolev emdedding; \(N\)-dimensional ball; infinite slab; rectilinear wave-guide PDFBibTeX XMLCite \textit{L. Brasco} et al., J. Funct. Anal. 274, No. 6, 1825--1863 (2018; Zbl 1388.35135) Full Text: DOI arXiv References: [1] Belloni, M.; Kawohl, B., A direct uniqueness proof for equations involving the \(p\)-Laplace operator, Manuscripta Math., 109, 229-231 (2002) · Zbl 1100.35032 [2] Brasco, L.; Franzina, G., Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Math. J., 37, 769-799 (2014) · Zbl 1315.47054 [3] Brasco, L.; Ruffini, B., Compact Sobolev embeddings and torsion functions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34, 4, 817-843 (2017) · Zbl 1378.46023 [4] Brezis, H.; Oswald, L., Remarks on sublinear elliptic equations, Nonlinear Anal., 10, 55-64 (1986) · Zbl 0593.35045 [5] Bucur, D.; Buttazzo, G., On the characterization of the compact embedding of Sobolev spaces, Calc. Var. Partial Differential Equations, 44, 455-475 (2012) · Zbl 1241.49023 [6] Bucur, D.; Buttazzo, G.; Velichkov, B., Spectral optimization problems for potentials and measures, SIAM J. Math. Anal., 46, 2956-2986 (2014) · Zbl 1301.49122 [7] Deny, J.; Lions, J.-L., Les espaces du type de Beppo Levi, Ann. Inst. Fourier, 5, 305-370 (1954) · Zbl 0065.09903 [8] Devyver, B.; Fraas, M.; Pinchover, Y., Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon, J. Funct. Anal., 266, 4422-4489 (2014) · Zbl 1298.47057 [9] Franzina, G.; Lamberti, P. D., Existence and uniqueness for a \(p\)-Laplacian nonlinear eigenvalue problem, Electron. J. Differential Equations, 26 (2010), 10 pp. · Zbl 1188.35125 [10] Lundholm, D., Geometric extensions of many-particle Hardy inequalities, J. Phys. A, 48, Article 175203 pp. (2015) · Zbl 1328.81267 [11] Maz’ya, V., Sobolev Spaces, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, vol. 342 (2011), Springer: Springer Heidelberg [12] Moser, J., On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math., 16, 577-591 (1961) · Zbl 0111.09302 [13] Opic, B.; Kufner, A., Hardy-Type Inequalities, Pitman Research Notes in Mathematics Series, vol. 219 (1990), Longman Scientific & Technical: Longman Scientific & Technical Harlow · Zbl 0698.26007 [14] Talenti, G., Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110, 353-372 (1976) · Zbl 0353.46018 [15] Teschl, G., Mathematical Methods in Quantum Mechanics. With Applications to Schrödinger Operators, Graduate Studies in Mathematics, vol. 157 (2014), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1342.81003 [16] van den Berg, M.; Bucur, D., On the torsion function with Robin or Dirichlet boundary conditions, J. Funct. Anal., 266, 1647-1666 (2014) · Zbl 1292.35141 [17] van den Berg, M.; Carroll, T., Hardy inequality and \(L^p\) estimates for the torsion function, Bull. Lond. Math. Soc., 41, 980-986 (2009) · Zbl 1180.35396 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.