Positive solutions of semilinear elliptic boundary value problems

  • Klaus Schmitt
Part of the NATO ASI Series book series (ASIC, volume 472)


The paper is concerned with existence questions for positive solutions (ground states) of boundary value problems for semilinear elliptic partial differential equations. Global continuation and bifurcation results are used to obtain the existence of unbounded solution continua whenever the nonlinear terms depend upon a real parameter. Results are presented for various classes of nonlinear terms which are classified depending on their asymptotic growth, such as linear, superlinear, subcritical, and supercritical growth. Results describing the influence of the geometry, topology and dimension of the domain on the solution structure are also discussed.


Solution Continuum Nonlinear Elliptic Equation Radial Solution Principal Eigenvalue Critical Sobolev Exponent 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [AS]
    W. Allegretto and D. Siegel, Picone’s identity and the moving plane procedure, to appear.Google Scholar
  2. [ANZ]
    W. Allegretto, P. Nistri, and P. Zecca, Positive solutions of elliptic nonpositone problems, Differential Integral Equations 5 (1992), 95–101.MathSciNetzbMATHGoogle Scholar
  3. [A1]
    H. Amann, Nonlinear eigenvalue problems having precisely two solutions, Math. Z. 150 (1976), 27–37.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [A2]
    H. Amann, Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z. 150 (1976), 281–295.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [AAM]
    H. Amann, A. Ambrosetti, and G. Mancini, Elliptic equations with noninvertible Fredholm linear part and bounded nonlinearities, Math. Z. 158 (1978), 179–194.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [AAB]
    A. Ambrosetti, D. Arcoya, and B. Buffoni, Positive solutions for some semipositone problems via bifurcation theory, Differential Integral Equations 7 (1994), 655–664.MathSciNetzbMATHGoogle Scholar
  7. [BC]
    A. Bahri and J. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain, Comm. Pure AppL Math. 41 (1988), 253–294.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [B]
    C. Bandle, Isoperimetric Inequalities and Applications, Pitman, Boston, London, 1980.zbMATHGoogle Scholar
  9. [BCM]
    C. Bandle, C. Coffman and M. Marcus, Nonlinear elliptic problems in annular domains, J. Differential Equations 69 (1987), 332–345.MathSciNetCrossRefGoogle Scholar
  10. [BK]
    C. Bandle and M. Kwong, Semilinear problems in annular domains, Z. Angew. Math. Phys. 40 (1989), 245–257.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [BE]
    J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, Springer, New York, 1989.zbMATHGoogle Scholar
  12. [BL]
    H. Brascamp and L. Lieb, On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log concave functions and with an application to a diffusion equation, J. Fund. Anal. 22 (1976), 366–389.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [Br1]
    H. Brezis, Elliptic equations with limiting Sobolev exponents — the impact of topology, Comm. Pure Appl. Math. 39 (1986), S17–S39.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [Br2]
    H. Brezis, Nonlinear elliptic equations involving the critical Sobolev exponent — survey and perspectives, in: Directions in Partial Differential Equations, Academic Press, New York, 1987, 17–36.Google Scholar
  15. [BN]
    H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [BuN]
    C. Budd and J. Norbury, Semilinear elliptic equations and supercritical growth, J. Differential Equations 68 (1987), 169–197.Google Scholar
  17. [CS]
    A. Castro and R. Shivaji, Nonnegative solutions for a class of radially symmetric nonpositone problems, Proc. Amer. Math. Soc. 106 (1989), 735–740.MathSciNetzbMATHGoogle Scholar
  18. [C]
    I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, New York, 1984.zbMATHGoogle Scholar
  19. [Ch]
    Y. Cheng, On the existence of radial solutions of a nonlinear elliptic boundary value problem in an annulus, Math. Nachr. 165 (1994), 61–77.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [CH]
    S. Chow and J. Hale, Methods of Bifurcation Theory, Springer, New York, 1982.zbMATHCrossRefGoogle Scholar
  21. [CM]
    C. V. Coffman and M. Marcus, Existence and uniqueness results for semilinear Dirichlet problems in annuli, Arch. Rational Mech. Anal. 108 (1991), 293–307.MathSciNetCrossRefGoogle Scholar
  22. [CJSS]
    D. Costa, H. Jeggle, R. Schaaf, and K. Schmitt, Oscillatory perturbations of linear problems at resonance, Results Math. 14 (1988), 275–287.MathSciNetzbMATHGoogle Scholar
  23. [CR]
    M. Crandall and P. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear eigenvalue problems, Arch. Rational Mech. Anal. 58 (1975), 207–218.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [D]
    E.N. Dancer, On the use of asymptotics in nonlinear boundary value problems, Ann. Mat. Pura Appl. 131 (1982), 167–185.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [DS]
    E. N. Dancer and K. Schmitt, On positive solutions of semilinear elliptic equations, Proc. Amer. Math. Soc. 101 (1987), 445–452.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [DaS]
    H. Dang and K. Schmitt, Existence of positive solutions for semilinear elliptic equations in annular domains, Differential Integral Equations 7 (1994), 747–758.MathSciNetzbMATHGoogle Scholar
  27. [D]
    K. Deimling, Nonlinear Analysis, Springer, New York, 1985.zbMATHGoogle Scholar
  28. [EH]
    L. Erbe and S. Hu, On the existence of multiple positive solutions of nonlinear boundary value problems, to appear.Google Scholar
  29. [EW]
    L. Erbe and H. Wang, Existence and nonexistence of positive solutions for elliptic equations in an annulus, to appear.Google Scholar
  30. [FLN]
    D. deFigueiredo, P. Lions and R. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. (9) 61 (1982), 41–63.MathSciNetGoogle Scholar
  31. [GNN]
    B. Gidas, W. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209–243.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [GT]
    D. Gilbarg and S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1983.zbMATHCrossRefGoogle Scholar
  33. [GS]
    G. Gustafson and K. Schmitt, Nonzero solutions of boundary value problems for second order ordinary and delay differential equations, J. Differential Equations 12 (1972), 125–147.MathSciNetCrossRefGoogle Scholar
  34. [HK]
    T. Healey and H. Kielhofer, Positivity of global branches of fully nonlinear elliptic boundary value problems, Proc. Amer. Math. Soc. 115 (1992), 1031–1036.MathSciNetzbMATHCrossRefGoogle Scholar
  35. [JS]
    W. Jager and K. Schmitt, Symmetry breaking in semilinear elliptic problems, in: Analysis, et cetera (P. Rabinowitz and E. Zehnder, eds.), Academic Press, New York, 1990, 451–470.Google Scholar
  36. [JL]
    D. Joseph and T. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal. 49 (1973), 241–269.MathSciNetzbMATHGoogle Scholar
  37. [KM]
    H. Kielhofer and S. Maier, Infinitely many positive solutions of semilinear elliptic problems via sub- and supersolutions, Comm. Partial Differential Equations to appear.Google Scholar
  38. [K1]
    M. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon, New York, 1964.Google Scholar
  39. [K2]
    M. Krasnosel’skii, Positive Solutions of Operator Equations, NoordhofT, Groningen, 1964.Google Scholar
  40. [Kw]
    M. Kwong, Uniqueness results for Emden — Fowler boundary value problems, Nonlinear Anal 16 (1991), 435–454.MathSciNetzbMATHCrossRefGoogle Scholar
  41. [LL]
    E. Landesman and A. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1970), 609–623.MathSciNetzbMATHGoogle Scholar
  42. [LL]
    M. Lee and S. Lin, Radially symmetric positive solutions and symmetry breaking for semipositone problems on balls, to appear.Google Scholar
  43. [L]
    C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on bounded domains, Comm. Partial Differential Equations 16 (1991), 491–526.MathSciNetzbMATHCrossRefGoogle Scholar
  44. [L1]
    S. Lin, On the existence of positive radial solutions for nonlinear elliptic equations in annular domains, J. Differential Equations 81 (1989), 221–233.MathSciNetzbMATHCrossRefGoogle Scholar
  45. [L2]
    S. Lin, Positive radial solutions and nonradial bifurcation for semilinear elliptic problems on annular domains, J. Differential Equations 86 (1990), 367–391.MathSciNetzbMATHCrossRefGoogle Scholar
  46. [L3]
    S. Lin, Existence of positive nonradial solutions for elliptic equations in annular domains, Trans. Amer. Math. Soc, to appear.Google Scholar
  47. [L4]
    S. Lin, Existence of many positive nonradial solutions for nonlinear elliptic equations on annulus, J. Differential Equations, to appear.Google Scholar
  48. [LP]
    S. S. Lin and F. M. Pai, Existence and multiplicity of positive radial solutions for semilinear elliptic equations in annular domains, SIAM J. Appl. Math. 22 (1991), 1500–1515.MathSciNetzbMATHCrossRefGoogle Scholar
  49. [Li]
    P. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), 441–467.MathSciNetzbMATHCrossRefGoogle Scholar
  50. [LS]
    D. Lupo and S. Solimini, A note on a resonance problem, Proc. Roy. Soc. Edinburgh Sect. A 102 (1986), 1–7.MathSciNetzbMATHCrossRefGoogle Scholar
  51. [MS]
    S. Maier and K. Schmitt, Asymptotic behavior of solution continua for semilinear elliptic problems, Canad. Appl. Math. Quart., to appear.Google Scholar
  52. [MSI]
    J. Mawhin and K. Schmitt, Landesman-Lazer type problems at an eigenvalue of odd multiplicity, Results Math. 14 (1988), 138–146.MathSciNetzbMATHGoogle Scholar
  53. [MS2]
    J. Mawhin and K. Schmitt, Nonlinear eigenvalue problems with the parameter near resonance, Ann. Polon. Math. 51 (1990), 241–248.MathSciNetzbMATHGoogle Scholar
  54. [Mc]
    J. McGough, On solution continua of supercritical quasilinear elliptic problems, Differential Integral Equations 7 (1994), 1453–1472.MathSciNetzbMATHGoogle Scholar
  55. [MP]
    F. Mignot and J. Puel, Solution radiale singuliere de -Δu = λe u, C. R. Acad. Sci. Paris 307 (1988), 379–382.MathSciNetzbMATHGoogle Scholar
  56. [NS1]
    K. Nagasaki and T. Suzuki, Radial and nonradial solutions for the nonlinear eigenvalue problem Δu + λe u = 0 on annuli in ℝ2, J. Differential Equations 87 (1990), 144–168.MathSciNetzbMATHCrossRefGoogle Scholar
  57. [NS2]
    K. Nagasaki and T. Suzuki, Radial solutions for Δu + λe u = 0 on annuli in higher dimension, J. Differential Equations 100 (1992), 137–161.MathSciNetzbMATHCrossRefGoogle Scholar
  58. [NN]
    W. Ni and R. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of Δu + f(u,r) = 0, Comm. Pure Appl Math. 38 (1985), 67–108.MathSciNetzbMATHCrossRefGoogle Scholar
  59. [OS]
    T. Ogawa and T. Suzuki, Nonlinear elliptic equations with critical growth related to the Trudinger inequality, to appear.Google Scholar
  60. [P]
    F. Pacard, Radial and non-radial solutions of -Δu = λf(u) on an annulus of ℝn, n ≥ 3, J. Differential Equations 101 (1993), 103–138.MathSciNetzbMATHCrossRefGoogle Scholar
  61. [PSS]
    H. Peitgen, D. Saupe and K. Schmitt, Nonlinear elliptic boundary value problems versus their finite difference approximations, J. Reine Angew. Math. 322 (1980), 75–117.MathSciNetGoogle Scholar
  62. [PS1]
    H. Peitgen and K. Schmitt, Perturbations globales topologiques des problemes non lineaires aux valeurs propres, C. R. Acad. Sci. Paris Ser. A 291 (1980), 271–274.MathSciNetzbMATHGoogle Scholar
  63. [PS2]
    H. Peitgen and K. Schmitt, Global topological perturbations of nonlinear elliptic eigenvalue problems, Math. Methods Appl. Sci. 5 (1983), 376–388.MathSciNetzbMATHCrossRefGoogle Scholar
  64. [PS3]
    H. Peitgen and K. Schmitt, Global analysis of two-parameter elliptic eigenvalue problems, Trans. Amer. Math. Soc. 283 (1984), 57–95.MathSciNetzbMATHCrossRefGoogle Scholar
  65. [Po]
    S. Pohozaev, Eigenfunctions of the equation Δu + λf(u) = 0, Sov. Math. Dokl. 6 (1965), 1408–1411.Google Scholar
  66. [PS]
    P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J. 35 (1986), 681–703.MathSciNetzbMATHCrossRefGoogle Scholar
  67. [R1]
    P. Rabinowitz, On bifurcation from infinity, J. Differential Equations 14 (1983), 462–475.MathSciNetCrossRefGoogle Scholar
  68. [R2]
    P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Ser. in Math. 65, Amer. Math. Soc, Providence, RI, 1986.Google Scholar
  69. [Re]
    F. Rellich, Darstellung der Eigenwerte von Δu + λu = 0 durch ein Randintegral, Math. Z. 46 (1940), 635–636.MathSciNetCrossRefGoogle Scholar
  70. [S]
    J. Santanilla, Existence and nonexistence of positive radial solutions for some semilinear elliptic problems in annular domains, Nonlinear Anal. 16 (1991), 861–877.MathSciNetzbMATHCrossRefGoogle Scholar
  71. [Sc]
    R. Schaaf, Uniqueness for semilinear elliptic problems; supercritical growth and domain geometry, to appear.Google Scholar
  72. [SSI]
    R. Schaaf and K. Schmitt, A class of nonlinear Sturm-Liouville problems with infinitely many solutions, Trans. Amer. Math. Soc. 306 (1988), 853–859.MathSciNetzbMATHCrossRefGoogle Scholar
  73. [SS2]
    R. Schaaf and K. Schmitt, Periodic perturbations of linear problems at resonance on convex domains, Rocky Mountain J. Math. 20 (1990), 1119–1131.MathSciNetzbMATHCrossRefGoogle Scholar
  74. [SS3]
    R. Schaaf and K. Schmitt, Oscillatory perturbations of linear problems at resonance: Some numerical experiments, in: Computational Solution of Nonlinear Systems of Equations (E. Allgower and K. Georg, eds.), Lectures in Appl. Math. 26, Amer. Math. Soc., Providence, RI, 1990, 541–559.Google Scholar
  75. [SS4]
    R. Schaaf and K. Schmitt, Asymptotic behavior of positive solution branches of elliptic problems with linear part at resonance, Z. Angew. Math. Phys. 43 (1992), 645–675.MathSciNetzbMATHCrossRefGoogle Scholar
  76. [S1]
    K. Schmitt, Boundary value problems for quasilinear second order elliptic equations, Nonlinear Anal. 2 (1978), 263–309.MathSciNetzbMATHCrossRefGoogle Scholar
  77. [S2]
    K. Schmitt, A Study of Eigenvalue and Bifurcation Problems for Nonlinear Elliptic Partial Differential Equations via Topological Continuation Methods, CABAY, Louvain-la-Neuve, 1982.Google Scholar
  78. [SW1]
    J. Smoller and A. Wasserman, Symmetry breaking for positive solutions of semilinear elliptic equations, Arch. Rational Mech. Anal. 95 (1986), 217–225.MathSciNetzbMATHGoogle Scholar
  79. [SW2]
    J. Smoller and A. Wasserman, Existence of positive solutions for semilinear elliptic equations in general domains, Arch. Rational Mech. Anal. 98 (1987), 229–249.MathSciNetzbMATHCrossRefGoogle Scholar
  80. [So]
    S. Solimini, On the solvability of some elliptic partial differential equations with the linear part at resonance, J. Math. Anal. Appl. 117 (1986), 138–152.MathSciNetzbMATHCrossRefGoogle Scholar
  81. [W]
    J. Ward, A boundary value problem with a periodic nonlinearity, Nonlinear Anal. 10 (1986), 207–213.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Klaus Schmitt
    • 1
  1. 1.Department of MathematicsUniversity of UtahSalt Lake CityUSA

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