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  • 1 Department of Mathematics and Statistics, University of Calgary Calgary AB, Canada T2N 1N4
  • | 2 Centre of Applied Mathematics, University of West Bohemia Univerzitní 22, 306 14 Plzeň, Czech Republic
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A version of Sturm--Liouville theory is given for the one-dimensional p-Laplacian including the radial case. The treatment is modern but follows the strategy of Elbert's early work. Topics include a Prüfer-type transformation, eigenvalue existence, asymptotics and variational principles, and eigenfunction oscillation.

  • Otani, M., A Remark on certain nonlinear elliptic equations, Proc. Fac. Sci. Tokai Univ.19 (1984), 23-28. MR86c:35009

    'A Remark on certain nonlinear elliptic equations ' () 19 Proc. Fac. Sci. Tokai Univ. : 23 -28.

    • Search Google Scholar
  • Naito, Y., Uniqueness of positive solutions of quasilinear differential equations, Diff. Int. Equations8 (1995), 1813-1822. MR96q:34039

    'Uniqueness of positive solutions of quasilinear differential equations ' () 8 Diff. Int. Equations : 1813 -1822.

    • Search Google Scholar
  • Delpino, M., Manásevich, R. and Murúa, A. E., Existence and multiplicity of solutions with prescribed period for a second order quasilinear ODE, Nonlinear AnalysisTMA18 (1992), 79-92. MR92m:34055

    'Existence and multiplicity of solutions with prescribed period for a second order quasilinear ODE ' () TMA18 Nonlinear Analysis : 79 -92.

    • Search Google Scholar
  • Drábek, P. and Manásevich, R., On the closed solution to some nonhomogeneous eigenvalue problems with p-Laplacian, J. Differential and Integral Equations12(6), (1999), 773-788. MR2000i:34036

    'On the closed solution to some nonhomogeneous eigenvalue problems with p-Laplacian ' () 12 J. Differential and Integral Equations : 773 -788.

    • Search Google Scholar
  • Drábek, P. and Robinson, S. B., Resonance problems for the p-Laplacian, J. Functional Analysis169 (1999), 189-200. MR2000j:35096

    'Resonance problems for the p-Laplacian ' () 169 J. Functional Analysis : 189 -200.

  • Eberhart, W. and Elbert, Á., On the eigenvalues of a half-linear boundary value problem, Math. Nachr.213 (2000), 57-76. MR2001b:34035

    'On the eigenvalues of a half-linear boundary value problem ' () 213 Math. Nachr. : 57 -76.

    • Search Google Scholar
  • Atkinson, F. V., Discrete and Continuous Boundary Value Problems, Academic Press (1964). MR31#416

    Discrete and Continuous Boundary Value Problems , ().

  • Beesack, P., Hardy's inequality and its extensions, Pacific J. Math.11 (1961), 39-61. MR22#12187

    'Hardy's inequality and its extensions ' () 11 Pacific J. Math. : 39 -61.

  • Beesack, P., Integral inequalities involving a function and its derivative, Amer. Math. Monthly78 (1971), 705-741. MR48#4235

    'Integral inequalities involving a function and its derivative ' () 78 Amer. Math. Monthly : 705 -741.

    • Search Google Scholar
  • Binding, P. A., Browne, P. J. and Seddighi, K., Sturm-Liouville problems with eigenparameter dependent boundary conditions, Proc. Roy. Soc. Edinburgh125A (1993), 3555-3559. MR95k:34039

    'Sturm-Liouville problems with eigenparameter dependent boundary conditions ' () 125A Proc. Roy. Soc. Edinburgh : 3555 -3559.

    • Search Google Scholar
  • Elbert, Á., Takasi, K. and Tinagawa, T., An Oscillatory half-linear differential equation, Arch. Math.33 (1997), 355-361. MR98m:34071

    'An Oscillatory half-linear differential equation ' () 33 Arch. Math. : 355 -361.

  • Farby, C. and Fayyad, D., Periodic solutions of second order differential equations with a p-Laplacian and asymmetric nonlinearities, Rend. Ist. Mat. Univ. Trieste24 (1992), 207-227.

    'Periodic solutions of second order differential equations with a p-Laplacian and asymmetric nonlinearities ' () 24 Rend. Ist. Mat. Univ. Trieste : 207 -227.

    • Search Google Scholar
  • Friedlander, L., Asymptotic behaviour of the eigenvalues of the p-Laplacian, Comm. P.D.E. (1989), 1959-1069.

    'Asymptotic behaviour of the eigenvalues of the p-Laplacian ' () Comm. P.D.E. : 1959 -1069.

    • Search Google Scholar
  • FuČíak, S., NeČS, J., SouČEk, J. and SouČEk, V., Spectral analysis of nonlinear operators, Springer Lecture Notes in Mathematics346 (1973). MR57#7280

    'Spectral analysis of nonlinear operators ' () 346 Springer Lecture Notes in Mathematics .

    • Search Google Scholar
  • Guedda, M. and Veron, L., Bifurcation phenomena associated to the p-Laplacian operator, Trans. Amer. Math. Soc.310 (1988), 419-431. MR89j:35024

    'Bifurcation phenomena associated to the p-Laplacian operator ' () 310 Trans. Amer. Math. Soc. : 419 -431.

    • Search Google Scholar
  • Ince, E. L., Ordinary Differential Equations, Dover reprint (1993).

  • Kusano, T. and Swanson, C. A., Radial entire solutions of a class of quasilinear elliptic equations, J. Differential Equations83 (1990), 379-399. MR91i:35013

    'Radial entire solutions of a class of quasilinear elliptic equations ' () 83 J. Differential Equations : 379 -399.

    • Search Google Scholar
  • García Azorero, P. and Peral Alonso, I., Comportment asymptotique des valeurs propres du p-Laplacian, C.R. Acad. Sci. Paris301 (1988), 75-78.

    'Comportment asymptotique des valeurs propres du p-Laplacian ' () 301 C.R. Acad. Sci. Paris : 75 -78.

    • Search Google Scholar
  • Ghoussoub, N., Duality and Perturbation Methods in Critical Point Theory, Cambridge University Press (1993). MR95a:58021

    Duality and Perturbation Methods in Critical Point Theory , ().

  • Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flow, 2nd ed., Gordon and Breach (New York, 1969). MR40#7610

    The Mathematical Theory of Viscous Incompressible Flow , ().

  • Lions, J. L., Quelques methodes de résolution des problémes aux limites non linéaires, Dunod (Paris, 1969). MR41#4326

    Quelques methodes de résolution des problémes aux limites non linéaires , ().

  • Lindqvist, P., Note on a nonlinear eigenvalue problem, Rocky Mount. J. Math.23 (1993), 281-288. MR94d:34031

    'Note on a nonlinear eigenvalue problem ' () 23 Rocky Mount. J. Math. : 281 -288.

  • Lindqvist, P., Some remarkable sine and cosine functions, Ricerche di Matematica44 (1995), 269-290. MR99q:33001

    'Some remarkable sine and cosine functions ' () 44 Ricerche di Matematica : 269 -290.

  • Li, H. J. and Yeh, C. C., Sturmian comparison theorem for half-linear second order differential equations, Proc. Royal Soc. Edinburgh125A (1995), 1193-1204.MR96i:34067

    'Sturmian comparison theorem for half-linear second order differential equations ' () 125A Proc. Royal Soc. Edinburgh : 1193 -1204.

    • Search Google Scholar
  • MaŘíak, R., Half-linear differential equations, PhD Thesis (Brno, 2000).

  • Reichel, W. and Walter, W., Sturm-Liouville type problems for the p-Laplacian under asymptotic nonresonance conditions, J. Differential Equations156 (1999), 50-70. MR2000e:34036

    'Sturm-Liouville type problems for the p-Laplacian under asymptotic nonresonance conditions ' () 156 J. Differential Equations : 50 -70.

    • Search Google Scholar
  • Szulkin, A., Ljusternik-Schnirelmannn theory on C1-manifolds, Ann. Inst. Henri Poincaré, Vol. 5, no. 2 (1988), 119-139. MR90a:58027

    'Ljusternik-Schnirelmannn theory on C1-manifolds ' () 5 Ann. Inst. Henri Poincaré : 119 -139.

    • Search Google Scholar
  • Takaşi, K., Ogata, A. and Usani, H., On the oscillation of solutions of second order quasilinear ordinary differential equations, Hiroshima Math. J.23 (1993), 645-667. MR95g:34053

    'On the oscillation of solutions of second order quasilinear ordinary differential equations ' () 23 Hiroshima Math. J. : 645 -667.

    • Search Google Scholar
  • Takaşi, K. and Yosida, N., Nonoscillation theorems for a class of quasilinear differential equations of second order, J. Math. Anal. Appl.189 (1995), 115-127. MR97f:34019

    'Nonoscillation theorems for a class of quasilinear differential equations of second order ' () 189 J. Math. Anal. Appl. : 115 -127.

    • Search Google Scholar
  • Walter, W., Sturm-Liouville theory for the radial Δp-operator, Math. Z.227 (1998), 175-185. MR99b:35073

    'Sturm-Liouville theory for the radial Δp-operator ' () 227 Math. Z. : 175 -185.

  • Binding, P. A., Drábek, P. and Huang, Y. X., On the Fredholm alternative for the p-Laplacian, Proc. Amer. Math. Society125 (1997), 3555-3559. MR98b:35058

    'On the Fredholm alternative for the p-Laplacian ' () 125 Proc. Amer. Math. Society : 3555 -3559.

    • Search Google Scholar
  • Binding, P. A. and Volkmer, H., Existence and asymptotics of eigenvalues of problems of Sturm-Liouville and Dirac type, J. Differential Equ.172 (2001), 116-133. MR2002d:34152

    'Existence and asymptotics of eigenvalues of problems of Sturm-Liouville and Dirac type ' () 172 J. Differential Equ. : 116 -133.

    • Search Google Scholar
  • Cuesta, M., On the Fučík Spectrum of the Laplacian and the p-Laplacian, in: Proceedings of Seminar in Differential Equations (P. Drábek, ed.), University of West Bohemia (Pilsen, 2000), pp. 67-96.

    'On the Fučík Spectrum of the Laplacian and the p-Laplacian ' , , .

  • Delpino, M. and Manásevich, R., Global bifurcation from the eigenvalues of the p-Laplacian, J. Differential Equations92 (1991), 226-251.

    'Global bifurcation from the eigenvalues of the p-Laplacian ' () 92 J. Differential Equations : 226 -251.

    • Search Google Scholar
  • Elbert, Á. A half-linear second order differential equation, in: Qualitative theory of differential equations, Coll. Math. Soc. J. Bolyai 30 (Szeged, 1979), 153-179. MR84g:34008

    'A half-linear second order differential equation, in: ' () 30 Qualitative theory of differential equations : 153 -179.

    • Search Google Scholar
  • Elbert, Á. The Wronskian and the half-linear differential equations, Studia Sci. Math. Hung.15 (1980), 101-105. MR84k:34013

    'The Wronskian and the half-linear differential equations ' () 15 Studia Sci. Math. Hung. : 101 -105.

    • Search Google Scholar
  • Elbert, Á. Qualitative properties of the half-linear second order differential equations, Communications of Computer and Automation Inst. of Hung. Acad. Sci.26 (1982), 27-33. MR85d:34036

    'Qualitative properties of the half-linear second order differential equations ' () 26 Communications of Computer and Automation Inst. of Hung. Acad. Sci. : 27 -33.

    • Search Google Scholar
  • Elbert, Á. Oscillation and nonoscillation theorems for some nonlinear ordinary differential equations, Springer Lecture Notes in Mathematics964 (1982), 187-212. MR84h:34056

    'Oscillation and nonoscillation theorems for some nonlinear ordinary differential equations ' () 964 Springer Lecture Notes in Mathematics : 187 -212.

    • Search Google Scholar
  • Pucci, P. and Serrin, J., Continuation and limit properties for solutions of strongly nonlinear second order differential equations, Asympt. Anal.4 (1991), 97-160. MR92j:34069

    'Continuation and limit properties for solutions of strongly nonlinear second order differential equations ' () 4 Asympt. Anal. : 97 -160.

    • Search Google Scholar
  • Rabinowitz, P. H., Some global results for nonlinear eigenvalue problems, J. Functional Analysis7 (1971), 487-513. MR46#745

    'Some global results for nonlinear eigenvalue problems ' () 7 J. Functional Analysis : 487 -513.

    • Search Google Scholar