Taiwanese Journal of Mathematics


G. Dwivedi and J. Tyagi

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In this article, we obtain several remarks on the qualitative questions such as Picone's identity, Morse index and Hardy-Rellich type inequality for biharmonic operators.

Article information

Taiwanese J. Math., Volume 19, Number 6 (2015), 1743-1758.

First available in Project Euclid: 4 July 2017

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Zentralblatt MATH identifier

Primary: 35J91: Semilinear elliptic equations with Laplacian, bi-Laplacian or poly- Laplacian
Secondary: 35B35: Stability

bi-Laplacian variational methods Morse index Hardy-Rellich inequality


Dwivedi, G.; Tyagi, J. REMARKS ON THE QUALITATIVE QUESTIONS FOR BIHARMONIC OPERATORS. Taiwanese J. Math. 19 (2015), no. 6, 1743--1758. doi:10.11650/tjm.19.2015.5566. https://projecteuclid.org/euclid.twjm/1499133737

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  • Adimurthi and S. Santra, Generalized Hardy-Rellich inequalities in critical dimension and its applications, Commun. Contemp. Math. 11(3) (2009), 367–394.
  • S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math. 12, 1959, 623–727.
  • W. Allegretto, Positive solutions and spectral properties of weakly coupled elliptic systems, J. Math. Anal. Appl. 120(2) (1986), 723–729.
  • ––––, On the principal eigenvalues of indefinite elliptic problems, Math. Z. 195(1) (1987), 29–35.
  • ––––, Sturmian theorems for second order systems, Proc. Amer. Math. Soc. 94(2) (1985), 291–296.
  • W. Allegretto and Y. X. Huang, A Picone's identity for the p-Laplacian and applications, Nonlinear Anal. 32(7) (1998), 819–830.
  • K. Bal, Generalized Picone's identity and its applications, Electron. J. Diff. Equations. 243 (2013), 1–6.
  • G. Bognár and O. Do\ptmrs šlý, Picone-type identity for pseudo p-Laplacian with variable power, Electron. J. Diff. Equations 174 (2012), 1–8.
  • H. Brézis and G. Stampacchia, Remarks on some fourth order variational inequalities, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4(2) (1977), 363–371.
  • D. R. Dunninger, A Picone integral identity for a class of fourth order elliptic differential inequalities, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 50(8) (1971), 630–641.
  • A. Ferrero, F. Gazzola and T. Weth, Positivity, symmetry and uniqueness for minimizers of second-order Sobolev inequalities, Ann. Mat. Pura Appl. 186(4) (2007), 565–578.
  • F. Gazzola, H. Grunau and G. Sweers, Polyharmonic boundary value problems, A monograph on positivity preserving and nonlinear higher order elliptic equations in bounded domain, Springer, 1991.
  • N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities, Math. Ann. 349(1) (2011), 1–57.
  • J. V. A. Goncalves, E. D. Silva and M. L. Silva, On positive solutions for a fourth order asymptotically linear elliptic equation under Navier boundary conditions, J. Math. Anal. Appl. 384 (2011), 387–399.
  • J. Jaro\ptmrs š, The higher-order Picone identity and comparison of half-linear differential equations of even order, Nonlinear Anal. 74(18) (2011), 7513–7518.
  • A. Manes and A. M. Micheletti, Un'estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine, Bollettino U.M.I. 7 1973, 285–301.
  • M. Picone, Un teorema sulle soluzioni delle equazioni lineari ellittiche autoaggiunte alle derivate parziali del secondo-ordine, Atti Accad. Naz. Lincei Rend. 20 (1911), 213–219.
  • M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Fourth Edition, Springer, 2007.
  • J. Tyagi, A nonlinear Picone's identity and its applications, Applied Mathematics Letters 26 (2013), 624–626.
  • R. C. A. M. Van der Vorst, Best constant for the embedding of the space $H^{2}(\Om)\cap H^{1}_{0}(\Om)$ into $L^{\frac{2N}{N-4}}(\Om)$, Diff. Int. Equations 6(2) (1993), 259–276.
  • N. Yoshida, Picone identities for half-linear elliptic operators with $p(x)$-Laplacians and applications to Sturmian comparison theory, Nonlinear Anal. 74 (2011), no. 16, 5631–5642.