Ground state alternative for p-Laplacian with potential term

  • Yehuda PinchoverEmail author
  • Kyril Tintarev


Let Ω be a domain in \(\mathbb{R}^d\), d  ≥  2, and 1 <  p  <  ∞. Fix \(V \in L_{\mathrm{loc}}^\infty(\Omega)\). Consider the functional Q and its Gâteaux derivative Q′ given by \( Q(u) := \mathop \int_\Omega (|\nabla u|^p+V|u|^p){\rm d}x,\,\, \frac{1}{p}Q^\prime (u) := -\nabla\cdot(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2}u.\) If Q  ≥  0 on\(C_0^{\infty}(\Omega)\), then either there is a positive continuous function W such that \(\int W|u|^p\,\mathrm{d}x\leq Q(u)\) for all\(u\in C_0^{\infty}(\Omega)\), or there is a sequence \(u_k\in C_0^{\infty}(\Omega)\) and a function v > 0 satisfying Q′ (v) = 0, such that Q(u k ) → 0, and \(u_k\to v\) in \(L^p_\mathrm{loc}(\Omega)\). In the latter case, v is (up to a multiplicative constant) the unique positive supersolution of the equation Q′ (u) = 0 in Ω, and one has for Q an inequality of Poincaré type: there exists a positive continuous function W such that for every \(\psi\in C_0^\infty(\Omega)\) satisfying \(\int \psi v\,{\rm d}x \neq 0\) there exists a constant C > 0 such that \(C^{-1}\int W|u|^p\,\mathrm{d}x\le Q(u)+C\left|\int u \psi\,\mathrm{d}x\right|^p\). As a consequence, we prove positivity properties for the quasilinear operator Q′ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.


Quasilinear elliptic operator p-Laplacian Ground state Positive solutions Green function Isolated singularity 

Mathematics Subject Classification (2000)

Primary 35J20 Secondary 35J60 35J70 49R50 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agmon, S. On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds. in:Methods of Functional Analysis and Theory of Elliptic Equations, (Naples, 1982), pp. 19–52. Liguori, Naples (1983)Google Scholar
  2. 2.
    Allegretto W., Huang Y.X. (1998) A Picone’s identity for the p-Laplacian and applications. Nonlinear Anal. 32, 819–830zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Allegretto W., Huang Y.X. (1999) Principal eigenvalues and Sturm comparison via Picone’s identity. J. Diff. Equ. 156, 427–438zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Barbatis G., Filippas S., Tertikas A. (2004) A unified approach to improved L p Hardy inequalities with best constants. Trans. Am. Math. Soc. 356, 2169–2196zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bidaut-Véron, M.-F., Borghol, R., Véron, L. Boundary Harnack inequality and a priori estimates of singular solutions of quasilinear elliptic equations. Calc. Var. Partial Diff. Equ. (to appear)(2006)Google Scholar
  6. 6.
    Brezis H., Lieb E.H. (1985) Sobolev inequalities with remainder terms. J. Funct. Anal. 62, 73–86zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Brezis H., Marcus M. (1997) Hardy’s inequalities revisited, Dedicated to Ennio De Giorgi. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25(4): 217–237zbMATHMathSciNetGoogle Scholar
  8. 8.
    Cycon H.L., Froese R.G., Kirsch W., Simon B. (1987) Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics. Springer, Berlin Heidelberg New YorkGoogle Scholar
  9. 9.
    del Pino M., Elgueta M., Manasevich R. (1989) A homotopic deformation along p of a Leray-Schauder degree result and existence for \((|u^{\prime}|^{p-2}u^{\prime})^{\prime}+f(t,u) = 0,\, u(0) = u(T) = 0,\, p > 1\). J. Diff. Equ. 80, 1–13zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    DiBenedetto E. (1983) C 1+α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7, 827–850zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Drábek P., Girg P., Takáč P., Ulm M. (2004) The Fredholm alternative for the p-Laplacian: bifurcation from infinity, existence and multiplicity. Indiana Univ. Math. J. 53, 433–482zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Drábek P., Kufner A., Nicolosi F. (1997) Quasilinear Elliptic Equations with Degenerations and Singularities, de Gruyter Series in Nonlinear Analysis and Applications, vol. 5, Walter de Gruyter & Co., BerlinGoogle Scholar
  13. 13.
    Filippas S., Tertikas A. (2002) Optimizing improved Hardy inequalities. J. Funct. Anal. 192, 186–233zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Fleckinger-Pellé, J., Hernández, J., Takáč, P., de Thélin, F. Uniqueness and positivity for solutions of equations with the p-Laplacian. In Proceedings of the Conference on Reaction-Diffusion Equations (Trieste, 1995), pp. 141–155. Lecture Notes in Pure and Applied Math. vol. 194. Marcel Dekker, New York (1998)Google Scholar
  15. 15.
    Fleckinger-Pellé J., Manásevich R.F., Stavrakakis N.M., de Thélin F. (1997) Principal eigenvalues for some quasilinear elliptic equations on \(\mathbb{R}^N\). Adv. Diff. Equ. 2, 981–1003Google Scholar
  16. 16.
    Fleckinger-Pellé J., Gossez J.-P., de Thélin F. (2004) Antimaximum principle in \(\mathbb{R}^N\): local versus global. J. Diff. Equ. 196, 119–133zbMATHCrossRefGoogle Scholar
  17. 17.
    García-Melián J., Sabina de Lis J. (1998) Maximum and comparison principles for operators involving the p-Laplacian. J. Math. Anal. Appl. 218, 49–65zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Gilbarg D., Serrin J. (1955/56) On isolated singularities of solutions of second order elliptic differential equations. J. Anal. Math. 4, 309–340MathSciNetGoogle Scholar
  19. 19.
    Guedda M., Véron L. (1988) Local and global properties of solutions of quasilinear elliptic equations. J. Diff. Equ. 76, 159–189zbMATHCrossRefGoogle Scholar
  20. 20.
    Hardt R., Hoffmann-Ostenhof M., Hoffmann-Ostenhof T., Nadirashvili N. (1999) Critical sets of solutions to elliptic equations. J. Diff. Geom. 51, 359–373zbMATHMathSciNetGoogle Scholar
  21. 21.
    Heinonen J., Kilpeläinen T., Martio O. (1993) Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs. Oxford University Press, New YorkGoogle Scholar
  22. 22.
    Jaroš J., Takaŝi K., Yoshida N. (2002) Picone-type inequalities for half-linear elliptic equations and their applications. Adv. Math. Sci. Appl. 12, 709–724MathSciNetzbMATHGoogle Scholar
  23. 23.
    Kichenassamy S., Véron L. (1986) Singular solutions of the p-Laplace equation. Math. Ann. 275, 599–615zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Klaus M., Simon B. (1979) Binding of Schrödinger particles through conspiracy of potential wells. Ann. Inst. Henri Poincaré, Sect. A Phys. Théor. 30, 83–87MathSciNetGoogle Scholar
  25. 25.
    Marcus M., Shafrir I. (2000) An eigenvalue problem related to Hardy’s L p inequality. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 29(4): 581–604zbMATHMathSciNetGoogle Scholar
  26. 26.
    Ovchinnikov Yu.N., Sigal I.M. (1979) Number of bound states of three-body systems and Efimov’s effect. Ann. Phys. 123, 274–295MathSciNetCrossRefGoogle Scholar
  27. 27.
    Pinchover Y. (1988) On positive solutions of second-order elliptic equations, stability results, and classification. Duke Math. J. 57, 955–980zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Pinchover Y. (1990) On criticality and ground states of second order elliptic equations, II. J. Diff. Equ. 87, 353–364zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Pinchover Y. (1995) On the localization of binding for Schrödinger operators and its extension to elliptic operators. J. Anal. Math. 66, 57–83zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Pinchover, Y.On principal eigenvalues for indefinite-weight elliptic problems. In: Spectral and Scattering Theory (Newark, DE, 1997), pp. 77–87. Plenum, New York (1998)Google Scholar
  31. 31.
    Pinchover Y., Tintarev K. (2005) Existence of minimizers for Schrödinger operators under domain perturbations with application to Hardy’s inequality. Indiana Univ. Math. J. 54, 1061–1074zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Pinchover Y., Tintarev K. (2006) Ground state alternative for singular Schrödinger operators. J. Funct. Analy. 230, 65–77zbMATHMathSciNetGoogle Scholar
  33. 33.
    Poliakovsky A., Shafrir I. (2005) Uniqueness of positive solutions for singular problems involving the p-Laplacian. Proc. Am. Math. Soc. 133, 2549–2557zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Serrin J. (1964) Local behavior of solutions of quasi-linear equations. Acta Math. 111, 247–302zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Serrin J. (1965) Isolated singularities of solutions of quasi-linear equations. Acta Math. 113, 219–240zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Simon B. (1980) Brownian motion, L p properties of Schrödinger operators and the localization of binding. J. Funct. Anal. 35, 215–229zbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Tolksdorf P. (1984) Regularity for a more general class of quasilinear elliptic equations, J. Diff. Equ. 51, 126–150zbMATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Véron L. (1996) Singularities of Solutions of Second Order Quasilinear Equations, Pitman Research Notes in Mathematics Series, vol. 353. Longman, HarlowzbMATHGoogle Scholar
  39. 39.
    Ziemer W.P. (1989) Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, Graduate Texts in Mathematics, vol. 120. Springer, Berlin Heidelberg New YorkzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of MathematicsTechnion - Israel Institute of TechnologyHaifaIsrael
  2. 2.Department of MathematicsUppsala UniversityUppsalaSweden

Personalised recommendations