Spectrum of Differential Operators

  • Dumitru Motreanu
  • Viorica Venera Motreanu
  • Nikolaos Papageorgiou


This chapter provides a self-contained account of the spectral properties of the following fundamental differential operators: Laplacian, p-Laplacian, and p-Laplacian plus an indefinite potential, with any 1 < p < +. The first section of the chapter examines the spectrum of the Laplacian separately under Dirichlet and Neumann boundary conditions, taking advantage of the essential feature that this refers to a linear operator. The second section addresses the spectrum of the p-Laplacian, again considering separately the Dirichlet and Neumann boundary conditions. Here the methods are completely different with respect to the Laplacian because the p-Laplacian is a nonlinear operator for p ≠ 2, making use of topological tools such as the Lyusternik–Schnirelmann principle. The third section extends this study to the more general class of nonlinear operators expressed as the sum of p-Laplacian and certain indefinite potential. Powerful related techniques are developed, for instance, the antimaximum principle, which is presented in a novel form. The fourth section addresses the Fučík spectrum, which incorporates the ordinary spectrum. The last section contains comments and information on relevant literature.


  1. 1.
    Abramovich, Y.A., Aliprantis, C.D.: An Invitation to Operator Theory. American Mathematical Society, Providence, RI (2002)MATHGoogle Scholar
  2. 2.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Elsevier/Academic, Amsterdam (2003)MATHGoogle Scholar
  3. 3.
    Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints. Mem. Amer. Math. Soc. 196, 915 (2008)MathSciNetGoogle Scholar
  4. 4.
    Aizicovici, S., Papageorgiou, N.S., Staicu, V.: The spectrum and an index formula for the Neumann p-Laplacian and multiple solutions for problems with a crossing nonlinearity. Discrete Contin. Dyn. Syst. 25, 431–456 (2009)MathSciNetMATHGoogle Scholar
  5. 5.
    Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Nonlinear resonant periodic problems with concave terms. J. Math. Anal. Appl. 375, 342–364 (2011)MathSciNetMATHGoogle Scholar
  6. 6.
    Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Positive solutions for nonlinear periodic problems with concave terms. J. Math. Anal. Appl. 381, 866–883 (2011)MathSciNetMATHGoogle Scholar
  7. 7.
    Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space. Dover Publications Inc., New York (1993)MATHGoogle Scholar
  8. 8.
    Allegretto, W., Huang, Y.X.: A Picone’s identity for the p-Laplacian and applications. Nonlin. Anal. 32, 819–830 (1998)MathSciNetMATHGoogle Scholar
  9. 9.
    Alves, C.O., Carrião, P.C., Miyagaki, O.H.: Multiple solutions for a problem with resonance involving the p-Laplacian. Abstr. Appl. Anal. 3, 191–201 (1998)MathSciNetMATHGoogle Scholar
  10. 10.
    Amann, H.: Saddle points and multiple solutions of differential equations. Math. Z. 169, 127–166 (1979)MathSciNetMATHGoogle Scholar
  11. 11.
    Amann, H.: A note on degree theory for gradient mappings. Proc. Amer. Math. Soc. 85, 591–595 (1982)MathSciNetMATHGoogle Scholar
  12. 12.
    Amann, H., Weiss, S.A.: On the uniqueness of the topological degree. Math. Z. 130, 39–54 (1973)MathSciNetMATHGoogle Scholar
  13. 13.
    Amann, H., Zehnder, E.: Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 7(4), 539–603 (1980)MathSciNetMATHGoogle Scholar
  14. 14.
    Ambrosetti, A., Arcoya, D.: An Introduction to Nonlinear Functional Analysis and Elliptic Problems. Birkhäuser, Boston (2011)MATHGoogle Scholar
  15. 15.
    Ambrosetti, A., Malchiodi, A.: Nonlinear Analysis and Semilinear Elliptic Problems. Cambridge University Press, Cambridge (2007)MATHGoogle Scholar
  16. 16.
    Ambrosetti, A., Prodi, G.: A Primer of Nonlinear Analysis. Cambridge University Press, Cambridge (1993)Google Scholar
  17. 17.
    Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)MathSciNetMATHGoogle Scholar
  18. 18.
    Ambrosetti, A., Brezis, H., Cerami, G.: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122, 519–543 (1994)MathSciNetMATHGoogle Scholar
  19. 19.
    Anane, A.: Simplicité et isolation de la première valeur propre du p-laplacien avec poids. C. R. Acad. Sci. Paris Sér. I Math. 305, 725–728 (1987)MathSciNetMATHGoogle Scholar
  20. 20.
    Anane, A., Tsouli, N.: On the second eigenvalue of the p-Laplacian. In: Nonlinear Partial Differential Equations (Fès, 1994). Longman, Harlow (1996)Google Scholar
  21. 21.
    Anello, G.: Existence of infinitely many weak solutions for a Neumann problem. Nonlin. Anal. 57, 199–209 (2004)MathSciNetMATHGoogle Scholar
  22. 22.
    Appell, J., Zabreĭko, P.P.: Nonlinear Superposition Operators. Cambridge University Press. Cambridge (1990)MATHGoogle Scholar
  23. 23.
    Arcoya, D., Carmona, J., Pellacci, B.: Bifurcation for some quasilinear operators. Proc. Roy. Soc. Edinb. Sect. A 131, 733–765 (2001)MathSciNetMATHGoogle Scholar
  24. 24.
    Arias, M., Campos, J.: Radial Fučik spectrum of the Laplace operator. J. Math. Anal. Appl. 190, 654–666 (1995)MathSciNetMATHGoogle Scholar
  25. 25.
    Arias, M., Campos, J., Cuesta, M., Gossez, J.-P.: Asymmetric elliptic problems with indefinite weights. Ann. Inst. H. Poincaré Anal. Non Linéaire 19, 581–616 (2002)MathSciNetMATHGoogle Scholar
  26. 26.
    Averna, D., Marano, S.A., Motreanu, D.: Multiple solutions for a Dirichlet problem with p-Laplacian and set-valued nonlinearity. Bull. Aust. Math. Soc. 77, 285–303 (2008)MathSciNetMATHGoogle Scholar
  27. 27.
    Badiale, M., Serra, E.: Semilinear Elliptic Equations for Beginners: Existence Results via the Variational Approach. Springer, London (2011)Google Scholar
  28. 28.
    Balanov, Z., Krawcewicz, W., Steinlein, H.: Applied Equivariant Degree. American Institute of Mathematical Sciences (AIMS), Springfield (2006)MATHGoogle Scholar
  29. 29.
    Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff International Publishing, Leiden (1976)MATHGoogle Scholar
  30. 30.
    Barbu, V.: Analysis and Control of Nonlinear Infinite-Dimensional Systems. Academic Inc., Boston (1993)MATHGoogle Scholar
  31. 31.
    Barletta, G., Papageorgiou, N.S.: A multiplicity theorem for the Neumann p-Laplacian with an asymmetric nonsmooth potential. J. Global Optim. 39, 365–392 (2007)MathSciNetMATHGoogle Scholar
  32. 32.
    Barletta, G., Papageorgiou, N.S.: Nonautonomous second order periodic systems: existence and multiplicity of solutions. J. Nonlin. Convex Anal. 8, 373–390 (2007)MathSciNetMATHGoogle Scholar
  33. 33.
    Bartolo, P., Benci, V., Fortunato, D.: Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity. Nonlin. Anal. 7, 981–1012 (1983)MathSciNetMATHGoogle Scholar
  34. 34.
    Bartsch, T.: Topological Methods for Variational Problems with Symmetries. Lecture Notes in Mathematics, vol. 1560. Springer, Berlin (1993)Google Scholar
  35. 35.
    Bartsch, T.: Infinitely many solutions of a symmetric Dirichlet problem. Nonlin. Anal. 20, 1205–1216 (1993)MathSciNetMATHGoogle Scholar
  36. 36.
    Bartsch, T., Li, S.: Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlin. Anal. 28, 419–441 (1997)MathSciNetMATHGoogle Scholar
  37. 37.
    Bartsch, T., Liu, Z.: On a superlinear elliptic p-Laplacian equation. J. Differ. Equat. 198, 149–175 (2004)MathSciNetMATHGoogle Scholar
  38. 38.
    Bartsch, T., Liu, Z., Weth, T.: Nodal solutions of a p-Laplacian equation. Proc. London Math. Soc. 91(3), 129–152 (2005)MathSciNetMATHGoogle Scholar
  39. 39.
    Benci, V.: On critical point theory for indefinite functionals in the presence of symmetries. Trans. Amer. Math. Soc. 274, 533–572 (1982)MathSciNetMATHGoogle Scholar
  40. 40.
    Benci, V., Rabinowitz, P.H.: Critical point theorems for indefinite functionals. Invent. Math. 52, 241–273 (1979)MathSciNetMATHGoogle Scholar
  41. 41.
    Ben-Naoum, A.K., De Coster, C.: On the existence and multiplicity of positive solutions of the p-Laplacian separated boundary value problem. Differ. Integr. Equat. 10, 1093–1112 (1997)MATHGoogle Scholar
  42. 42.
    Benyamini, Y., Sternfeld, Y.: Spheres in infinite-dimensional normed spaces are Lipschitz contractible. Proc. Amer. Math. Soc. 88, 439–445 (1983)MathSciNetMATHGoogle Scholar
  43. 43.
    Bessaga, C.: Every infinite-dimensional Hilbert space is diffeomorphic with its unit sphere. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 14, 27–31 (1966)MathSciNetMATHGoogle Scholar
  44. 44.
    Beurling, A., Livingston, A.E.: A theorem on duality mappings in Banach spaces. Ark. Mat. 4, 405–411 (1962)MathSciNetMATHGoogle Scholar
  45. 45.
    Binding, P.A., Rynne, B.P.: The spectrum of the periodic p-Laplacian. J. Differ. Equat. 235, 199–218 (2007)MathSciNetMATHGoogle Scholar
  46. 46.
    Binding, P.A., Rynne, B.P.: Variational and non-variational eigenvalues of the p-Laplacian. J. Differ. Equat. 244, 24–39 (2008)MathSciNetMATHGoogle Scholar
  47. 47.
    Blanchard, P., Brüning, E.: Variational Methods in Mathematical Physics. Springer, Berlin (1992)MATHGoogle Scholar
  48. 48.
    Bonanno, G., Candito, P.: Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian. Arch. Math. (Basel) 80, 424–429 (2003)MathSciNetMATHGoogle Scholar
  49. 49.
    Borwein, J.M., Vanderwerff, J.D.: Convex Functions: Constructions, Characterizations and Counterexamples. Cambridge University Press, Cambridge (2010)Google Scholar
  50. 50.
    Brezis, H.: Équations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier (Grenoble) 18, 115–175 (1968)MathSciNetMATHGoogle Scholar
  51. 51.
    Brézis, H.: Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert. North-Holland Publishing Co., Amsterdam (1973)MATHGoogle Scholar
  52. 52.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)MATHGoogle Scholar
  53. 53.
    Brezis, H., Nirenberg, L.: Remarks on finding critical points. Comm. Pure Appl. Math. 44, 939–963 (1991)MathSciNetMATHGoogle Scholar
  54. 54.
    Brezis, H., Nirenberg, L.: H 1 versus C 1 local minimizers. C. R. Acad. Sci. Paris Sér. I Math. 317, 465–472 (1993)MathSciNetMATHGoogle Scholar
  55. 55.
    Brøndsted, A., Rockafellar, R.T.: On the subdifferentiability of convex functions. Proc. Amer. Math. Soc. 16, 605–611 (1965)MathSciNetGoogle Scholar
  56. 56.
    Brouwer, L.E.J.: Über Abbildung von Mannigfaltigkeiten. Math. Ann. 71, 97–115 (1912)MATHGoogle Scholar
  57. 57.
    Browder, F.E.: Nonlinear maximal monotone operators in Banach space. Math. Ann. 175, 89–113 (1968)MathSciNetMATHGoogle Scholar
  58. 58.
    Browder, F.E.: Nonlinear monotone and accretive operators in Banach spaces. Proc. Nat. Acad. Sci. U.S.A. 61, 388–393 (1968)MathSciNetMATHGoogle Scholar
  59. 59.
    Browder, F.E.: Nonlinear operators and nonlinear equations of evolution in Banach spaces. In: Nonlinear Functional Analysis (Proc. Sympos. Pure Math., vol. XVIII, Part 2, Chicago, Ill., 1968), pp. 1–308. American Mathematical Society, Providence (1976)Google Scholar
  60. 60.
    Browder, F.E.: Fixed point theory and nonlinear problems. Bull. Amer. Math. Soc. (N.S.) 9, 1–39 (1983)Google Scholar
  61. 61.
    Browder, F.E.: Degree of mapping for nonlinear mappings of monotone type. Proc. Nat. Acad. Sci. USA 80, 1771–1773 (1983)MathSciNetMATHGoogle Scholar
  62. 62.
    Browder, F.E., Hess, P.: Nonlinear mappings of monotone type in Banach spaces. J. Funct. Anal. 11, 251–294 (1972)MathSciNetMATHGoogle Scholar
  63. 63.
    Các, N.P.: On nontrivial solutions of a Dirichlet problem whose jumping nonlinearity crosses a multiple eigenvalue. J. Differ. Equat. 80, 379–404 (1989)MATHGoogle Scholar
  64. 64.
    Čaklović, L., Li, S.J., Willem, M.: A note on Palais-Smale condition and coercivity. Differ. Integr. Equat. 3, 799–800 (1990)MATHGoogle Scholar
  65. 65.
    Cambini, A.: Sul lemma di M. Morse. Boll. Un. Mat. Ital. 7(4), 87–93 (1973)MathSciNetMATHGoogle Scholar
  66. 66.
    Candito, P., Livrea, R., Motreanu, D.: \(\mathbb{Z}_{2}\)-symmetric critical point theorems for non-differentiable functions. Glasg. Math. J. 50, 447–466 (2008)Google Scholar
  67. 67.
    Candito, P., Livrea, R., Motreanu, D.: Bounded Palais-Smale sequences for non-differentiable functions. Nonlin. Anal.74, 5446–5454 (2011)MathSciNetMATHGoogle Scholar
  68. 68.
    Carl, S., Motreanu, D.: Quasilinear elliptic inclusions of hemivariational type: extremality and compactness of the solution set. J. Math. Anal. Appl. 286, 147–159 (2003)MathSciNetMATHGoogle Scholar
  69. 69.
    Carl, S., Motreanu, D.: Constant-sign and sign-changing solutions of a nonlinear eigenvalue problem involving the p-Laplacian. Differ. Integr. Equat. 20, 309–324 (2007)MathSciNetMATHGoogle Scholar
  70. 70.
    Carl, S., Motreanu, D.: Constant-sign and sign-changing solutions for nonlinear eigenvalue problems. Nonlin. Anal. 68, 2668–2676 (2008)MathSciNetMATHGoogle Scholar
  71. 71.
    Carl, S., Motreanu, D.: Multiple and sign-changing solutions for the multivalued p-Laplacian equation. Math. Nachr. 283, 965–981 (2010)MathSciNetMATHGoogle Scholar
  72. 72.
    Carl, S., Le, V.K., Motreanu, D.: Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications. Springer, New York (2007)MATHGoogle Scholar
  73. 73.
    Casas, E., Fernández, L.A.: A Green’s formula for quasilinear elliptic operators. J. Math. Anal. Appl. 142, 62–73 (1989)MathSciNetMATHGoogle Scholar
  74. 74.
    Castro, A., Lazer, A.C.: Critical point theory and the number of solutions of a nonlinear Dirichlet problem. Ann. Mat. Pura Appl. 120(4), 113–137 (1979)MathSciNetMATHGoogle Scholar
  75. 75.
    Cerami, G.: An existence criterion for the critical points on unbounded manifolds. Istit. Lombardo Accad. Sci. Lett. Rend. A 112, 332–336 (1978)MathSciNetMATHGoogle Scholar
  76. 76.
    Chang, K.-C.: Solutions of asymptotically linear operator equations via Morse theory. Comm. Pure Appl. Math. 34, 693–712 (1981)MathSciNetMATHGoogle Scholar
  77. 77.
    Chang, K.-C.: Variational methods for nondifferentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80, 102–129 (1981)MathSciNetMATHGoogle Scholar
  78. 78.
    Chang, K.-C.: Infinite-Dimensional Morse Theory and Multiple Solution Problems. Birkhäuser, Boston (1993)MATHGoogle Scholar
  79. 79.
    Chang, K.-C.: H 1 versus C 1 isolated critical points. C. R. Acad. Sci. Paris Sér. I Math. 319, 441–446 (1994)MATHGoogle Scholar
  80. 80.
    Cherfils, L., Il’yasov, Y.: On the stationary solutions of generalized reaction diffusion equations with p & q-Laplacian. Commun. Pure Appl. Anal. 4, 9–22 (2005)MathSciNetMATHGoogle Scholar
  81. 81.
    Christensen, J.P.R.: Topology and Borel Structure. North-Holland Publishing Co., Amsterdam (1974)MATHGoogle Scholar
  82. 82.
    Ciorănescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic Publishers, Dordrecht (1990)MATHGoogle Scholar
  83. 83.
    Clark, D.C.: A variant of the Lusternik-Schnirelman theory. Indiana Univ. Math. J. 22, 65–74 (1972/1973)Google Scholar
  84. 84.
    Clarke, F.H.: Generalized gradients and applications. Trans. Amer. Math. Soc. 205, 247–262 (1975)MathSciNetMATHGoogle Scholar
  85. 85.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. A Wiley-Interscience Publication, Wiley Inc., New York (1983)Google Scholar
  86. 86.
    Coffman, C.V.: A minimum-maximum principle for a class of non-linear integral equations. J. Anal. Math. 22, 391–419 (1969)MathSciNetMATHGoogle Scholar
  87. 87.
    Cordaro, G.: Three periodic solutions to an eigenvalue problem for a class of second-order Hamiltonian systems. Abstr. Appl. Anal. 2003, 1037–1045 (2003)MathSciNetMATHGoogle Scholar
  88. 88.
    Corvellec, J.-N., Motreanu, V.V., Saccon, C.: Doubly resonant semilinear elliptic problems via nonsmooth critical point theory. J. Differ. Equat. 248, 2064–2091 (2010)MathSciNetMATHGoogle Scholar
  89. 89.
    Costa, D.G.: An Invitation to Variational Methods in Differential Equations. Birkhäuser, Boston (2007)MATHGoogle Scholar
  90. 90.
    Costa, D.G., Magalhães, C.A.: Existence results for perturbations of the p-Laplacian. Nonlin. Anal. 24, 409–418 (1995)MATHGoogle Scholar
  91. 91.
    Costa, D.G., Silva, E.A.: The Palais-Smale condition versus coercivity. Nonlin. Anal. 16, 371–381 (1991)MathSciNetMATHGoogle Scholar
  92. 92.
    Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. I. Interscience Publishers, New York (1953)Google Scholar
  93. 93.
    Crandall, M.G., Rabinowitz, P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971)MathSciNetMATHGoogle Scholar
  94. 94.
    Cuesta, M.: Minimax theorems on C 1 manifolds via Ekeland variational principle. Abstr. Appl. Anal. 2003, 757–768 (2003)MathSciNetMATHGoogle Scholar
  95. 95.
    Cuesta, M., Ramos Quoirin, H.: A weighted eigenvalue problem for the p-Laplacian plus a potential. NoDEA - Nonlin. Differ. Equat. Appl. 16, 469–491 (2009)MathSciNetMATHGoogle Scholar
  96. 96.
    Cuesta, M., Takáč, P.: A strong comparison principle for positive solutions of degenerate elliptic equations. Differ. Integr. Equat. 13, 721–746 (2000)MATHGoogle Scholar
  97. 97.
    Cuesta, M., Takáč, P.: Nonlinear eigenvalue problems for degenerate elliptic systems. Differ. Integr. Equat. 23, 1117–1138 (2010)MATHGoogle Scholar
  98. 98.
    Cuesta, M., de Figueiredo, D., Gossez, J.-P.: The beginning of the Fučik spectrum for the p-Laplacian. J. Differ. Equat. 159, 212–238 (1999)MATHGoogle Scholar
  99. 99.
    Damascelli, L.: Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results. Ann. Inst. H. Poincaré Anal. Non Linéaire 15, 493–516 (1998)MathSciNetMATHGoogle Scholar
  100. 100.
    Damascelli, L., Sciunzi, B.: Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations. Calc. Var. Partial Differ. Equat. 25, 139–159 (2006)MathSciNetMATHGoogle Scholar
  101. 101.
    Dancer, E.N.: Remarks on jumping nonlinearities. In: Topics in Nonlinear Analysis, pp. 101–116. Birkhäuser, Basel (1999)Google Scholar
  102. 102.
    Dancer, E.N., Du, Y.: On sign-changing solutions of certain semilinear elliptic problems. Appl. Anal. 56, 193–206 (1995)MathSciNetMATHGoogle Scholar
  103. 103.
    Day, M.M.: Some more uniformly convex spaces. Bull. Amer. Math. Soc. 47, 504–507 (1941)MathSciNetGoogle Scholar
  104. 104.
    de Figueiredo, D.G.: Positive solutions of semilinear elliptic problems. In: Differential Equations (Sao Paulo, 1981). Lecture Notes in Mathematics, vol. 957, pp. 34–87. Springer, Berlin (1982)Google Scholar
  105. 105.
    de Figueiredo, D.G.: Lectures on the Ekeland Variational Principle with Applications and Detours. Tata Institute of Fundamental Research, Bombay (1989)MATHGoogle Scholar
  106. 106.
    de Figueiredo, D.G., Gossez, J.-P.: On the first curve of the Fučik spectrum of an elliptic operator. Differ. Integr. Equat. 7, 1285–1302 (1994)MATHGoogle Scholar
  107. 107.
    Degiovanni, M., Marzocchi, M.: A critical point theory for nonsmooth functionals. Ann. Mat. Pura Appl. 167(4), 73–100 (1994)MathSciNetMATHGoogle Scholar
  108. 108.
    Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)MATHGoogle Scholar
  109. 109.
    Del Pezzo, L.M., Fernández Bonder, J.: An optimization problem for the first weighted eigenvalue problem plus a potential. Proc. Amer. Math. Soc. 138, 3551–3567 (2010)MathSciNetMATHGoogle Scholar
  110. 110.
    del Pino, M.A., Manásevich, R.F.: Global bifurcation from the eigenvalues of the p-Laplacian. J. Differ. Equat. 92, 226–251 (1991)MATHGoogle Scholar
  111. 111.
    del Pino, M.A., Manásevich, R.F., Murúa, A.E.: Existence and multiplicity of solutions with prescribed period for a second order quasilinear ODE. Nonlin. Anal. 18, 79–92 (1992)MATHGoogle Scholar
  112. 112.
    De Nápoli, P., Mariani, M.C.: Mountain pass solutions to equations of p-Laplacian type. Nonlin. Anal. 54, 1205–1219 (2003)MATHGoogle Scholar
  113. 113.
    Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Theory. Kluwer Academic Publishers, Boston (2003)Google Scholar
  114. 114.
    Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Applications. Kluwer Academic Publishers, Boston (2003)Google Scholar
  115. 115.
    Deny, J., Lions, J. L.: Les espaces du type de Beppo Levi. Ann. Inst. Fourier Grenoble 5, 305–370 (1953–54)Google Scholar
  116. 116.
    de Paiva, F.O., do Ó, J.M., de Medeiros, E.S.: Multiplicity results for some quasilinear elliptic problems. Topol. Methods Nonlin. Anal. 34, 77–89 (2009)Google Scholar
  117. 117.
    Deville, R., Godefroy, G., Zizler, V.: Smoothness and Renormings in Banach Spaces. Longman Scientific and Technical, Harlow (1993)MATHGoogle Scholar
  118. 118.
    DiBenedetto, E.: C 1+α local regularity of weak solutions of degenerate elliptic equations. Nonlin. Anal. 7, 827–850 (1983)MathSciNetMATHGoogle Scholar
  119. 119.
    Dold, A.: Lectures on Algebraic Topology. Springer, Berlin (1980)MATHGoogle Scholar
  120. 120.
    Drábek, P.: On the global bifurcation for a class of degenerate equations. Ann. Mat. Pura Appl. 159(4), 1–16 (1991)MathSciNetMATHGoogle Scholar
  121. 121.
    Drábek, P., Manásevich, R.: On the closed solution to some nonhomogeneous eigenvalue problems with p-Laplacian. Differ. Integr. Equat. 12, 773–788 (1999)MATHGoogle Scholar
  122. 122.
    Dugundji, J.: An extension of Tietze’s theorem. Pacific J. Math. 1, 353–367 (1951)MathSciNetMATHGoogle Scholar
  123. 123.
    Dugundji, J.: Topology. Allyn and Bacon Inc., Boston (1966)MATHGoogle Scholar
  124. 124.
    Dunford, N., Schwartz, J.T.: Linear Operators I: General Theory. Interscience Publishers, Inc., New York (1958)MATHGoogle Scholar
  125. 125.
    Eilenberg, S., Steenrod, N.: Foundations of Algebraic Topology. Princeton University Press, Princeton (1952)MATHGoogle Scholar
  126. 126.
    Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)MathSciNetMATHGoogle Scholar
  127. 127.
    Ekeland, I.: Nonconvex minimization problems. Bull. Amer. Math. Soc. (N.S.) 1, 443–474 (1979)Google Scholar
  128. 128.
    Ekeland, I.: Convexity Methods in Hamiltonian Mechanics. Springer, Berlin (1990)MATHGoogle Scholar
  129. 129.
    Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland Publishing Co., Amsterdam (1976)MATHGoogle Scholar
  130. 130.
    Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (1998)MATHGoogle Scholar
  131. 131.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC, Boca Raton (1992)MATHGoogle Scholar
  132. 132.
    Fabry, C., Fonda, A.: Periodic solutions of nonlinear differential equations with double resonance. Ann. Mat. Pura Appl. 157(4), 99–116 (1990)MathSciNetMATHGoogle Scholar
  133. 133.
    Fadell, E.R., Rabinowitz, P.H.: Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math. 45, 139–174 (1978)MathSciNetMATHGoogle Scholar
  134. 134.
    Fadell, E.R., Husseini, S.Y., Rabinowitz, P.H.: Borsuk-Ulam theorems for arbitrary S 1 actions and applications. Trans. Amer. Math. Soc. 274, 345–360 (1982)MathSciNetMATHGoogle Scholar
  135. 135.
    Faraci, F.: Three periodic solutions for a second order nonautonomous system. J. Nonlin. Convex Anal. 3, 393–399 (2002)MathSciNetMATHGoogle Scholar
  136. 136.
    Faria, L.F.O., Miyagaki, O.H., Motreanu, D.: Comparison and positive solutions for problems with (p, q)-Laplacian and convection term. Proc. Edinb. Math. Soc. (2) (to appear)Google Scholar
  137. 137.
    Fernández Bonder, J., Del Pezzo, L.M.: An optimization problem for the first eigenvalue of the p-Laplacian plus a potential. Commun. Pure Appl. Anal. 5, 675–690 (2006)MathSciNetMATHGoogle Scholar
  138. 138.
    Filippakis, M.E., Papageorgiou, N.S.: Solutions for nonlinear variational inequalities with a nonsmooth potential. Abstr. Appl. Anal. 8, 635–649 (2004)MathSciNetGoogle Scholar
  139. 139.
    Finn, R., Gilbarg, D.: Asymptotic behavior and uniqueness of plane subsonic flows. Comm. Pure Appl. Math. 10, 23–63 (1957)MathSciNetMATHGoogle Scholar
  140. 140.
    Floret, K.: Weakly Compact Set. Lecture Notes in Mathematics, vol. 801. Springer, Berlin (1980)Google Scholar
  141. 141.
    Fonseca, I., Gangbo, W.: Degree Theory in Analysis and Applications. The Clarendon Press and Oxford University Press, New York (1995)MATHGoogle Scholar
  142. 142.
    Fredholm, J.: Sur une classe d’équations fonctionnelles. Acta Math. 27, 365–390 (1903)MathSciNetMATHGoogle Scholar
  143. 143.
    Fučík, S.: Boundary value problems with jumping nonlinearities. Časopis Pěst. Mat. 101, 69–87 (1976)MATHGoogle Scholar
  144. 144.
    Führer, L.: Ein elementarer analytischer Beweis zur Eindeutigkeit des Abbildungsgrades im Rn. Math. Nachr. 54, 259–267 (1972)MathSciNetMATHGoogle Scholar
  145. 145.
    Gagliardo, E.: Proprietà di alcune classi di funzioni in più variabili. Ricerche Mat. 7, 102–137 (1958)MathSciNetMATHGoogle Scholar
  146. 146.
    Gallouët, T., Kavian, O.: Résultats d’existence et de non-existence pour certains problèmes demi-linéaires à l’infini. Ann. Fac. Sci. Toulouse Math. 3(5), 201–246 (1981)MathSciNetMATHGoogle Scholar
  147. 147.
    García Azorero, J.P., Manfredi, J.J., Peral Alonso, I.: Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations. Commun. Contemp. Math. 2, 385–404 (2000)MathSciNetMATHGoogle Scholar
  148. 148.
    Garofalo, N., Lin, F.-H.: Unique continuation for elliptic operators: a geometric-variational approach. Comm. Pure Appl. Math. 40, 347–366 (1987)MathSciNetMATHGoogle Scholar
  149. 149.
    Gasiński, L.: Positive solutions for resonant boundary value problems with the scalar p-Laplacian and nonsmooth potential. Discrete Contin. Dyn. Syst. 17, 143–158 (2007)MathSciNetMATHGoogle Scholar
  150. 150.
    Gasiński, L., Papageorgiou, N.S.: Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems. Chapman and Hall/CRC, Boca Raton (2005)MATHGoogle Scholar
  151. 151.
    Gasiński, L., Papageorgiou, N.S.: Nonlinear Analysis. Chapman and Hall/CRC, Boca Raton, FL (2006)MATHGoogle Scholar
  152. 152.
    Gasiński, L., Papageorgiou, N.S.: Three nontrivial solutions for periodic problems with the p-Laplacian and a p-superlinear nonlinearity. Commun. Pure Appl. Anal. 8, 1421–1437 (2009)MathSciNetMATHGoogle Scholar
  153. 153.
    Gasiński, L., Papageorgiou, N.S.: Nodal and multiple constant sign solutions for resonant p-Laplacian equations with a nonsmooth potential. Nonlin. Anal. 71, 5747–5772 (2009)MATHGoogle Scholar
  154. 154.
    Gasiński, L., Papageorgiou, N.S.: Multiple solutions for asymptotically (p − 1)-homogeneous p-Laplacian equations. J. Funct. Anal. 262, 2403–2435 (2012)MathSciNetMATHGoogle Scholar
  155. 155.
    Gasiński, L., Papageorgiou, N.S.: Bifurcation-type results for nonlinear parametric elliptic equations. Proc. Roy. Soc. Edinb. Sect. A 142, 595–623 (2012)Google Scholar
  156. 156.
    Ghoussoub, N.: Duality and Perturbation Methods in Critical Point Theory. Cambridge University Press, Cambridge (1993)MATHGoogle Scholar
  157. 157.
    Ghoussoub, N., Preiss, D.: A general mountain pass principle for locating and classifying critical points. Ann. Inst. H. Poincaré Anal. Non Linéaire 6, 321–330 (1989)MathSciNetMATHGoogle Scholar
  158. 158.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)MATHGoogle Scholar
  159. 159.
    Giles, J.R.: Convex Analysis with Application in the Differentiation of Convex Functions. Pitman, Boston (1982)MATHGoogle Scholar
  160. 160.
    Godoy, T., Gossez, J.-P., Paczka, S.: On the antimaximum principle for the p-Laplacian with indefinite weight. Nonlin. Anal. 51, 449–467 (2002)MathSciNetMATHGoogle Scholar
  161. 161.
    Goeleven, D., Motreanu, D.: A degree-theoretic approach for the study of eigenvalue problems in variational-hemivariational inequalities. Differ. Integr. Equat. 10, 893–904 (1997)MathSciNetMATHGoogle Scholar
  162. 162.
    Goeleven, D., Motreanu, D., Panagiotopoulos, P.D.: Eigenvalue problems for variational-hemivariational inequalities at resonance. Nonlin. Anal. 33, 161–180 (1998)MathSciNetMATHGoogle Scholar
  163. 163.
    Gossez, J.-P., Omari, P.: Periodic solutions of a second order ordinary differential equation: a necessary and sufficient condition for nonresonance. J. Differ. Equat. 94, 67–82 (1991)MathSciNetMATHGoogle Scholar
  164. 164.
    Gossez, J.-P., Omari, P.: A necessary and sufficient condition of nonresonance for a semilinear Neumann problem. Proc. Amer. Math. Soc. 114, 433–442 (1992)MathSciNetMATHGoogle Scholar
  165. 165.
    Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003)MATHGoogle Scholar
  166. 166.
    Gromoll, D., Meyer, W.: On differentiable functions with isolated critical points. Topology 8, 361–369 (1969)MathSciNetMATHGoogle Scholar
  167. 167.
    Guedda, M., Véron, L.: Quasilinear elliptic equations involving critical Sobolev exponents. Nonlin. Anal. 13, 879–902 (1989)MATHGoogle Scholar
  168. 168.
    Guo, Z., Zhang, Z.: W 1, p versus C 1 local minimizers and multiplicity results for quasilinear elliptic equations. J. Math. Anal. Appl. 286, 32–50 (2003)MathSciNetMATHGoogle Scholar
  169. 169.
    Halmos, P.R.: Introduction to Hilbert Space and the Theory of Spectral Multiplicity. AMS Chelsea Publishing, Providence (1998)MATHGoogle Scholar
  170. 170.
    Heinz, E.: An elementary analytic theory of the degree of mapping in n-dimensional space. J. Math. Mech. 8, 231–247 (1959)MathSciNetMATHGoogle Scholar
  171. 171.
    Hilbert, D.: Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. IV. Gött. Nachr. 1906, 157–227 (1906)MATHGoogle Scholar
  172. 172.
    Hofer, H.: Variational and topological methods in partially ordered Hilbert spaces. Math. Ann. 261, 493–514 (1982)MathSciNetMATHGoogle Scholar
  173. 173.
    Hofer, H.: A geometric description of the neighbourhood of a critical point given by the mountain-pass theorem. J. London Math. Soc. 31(2), 566–570 (1985)MathSciNetMATHGoogle Scholar
  174. 174.
    Hu, S., Papageorgiou, N.S.: Generalizations of Browder’s degree theory. Trans. Amer. Math. Soc. 347, 233–259 (1995)MathSciNetMATHGoogle Scholar
  175. 175.
    Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis. vol. I. Theory. Kluwer Academic Publishers, Dordrecht (1997)MATHGoogle Scholar
  176. 176.
    Hu, S., Papageorgiou, N.S.: Multiple positive solutions for nonlinear eigenvalue problems with the p-Laplacian. Nonlin. Anal. 69, 4286–4300 (2008)MathSciNetMATHGoogle Scholar
  177. 177.
    Hu, S., Papageorgiou, N.S.: Nontrivial solutions for superquadratic nonautonomous periodic systems. Topol. Methods Nonlin. Anal. 34, 327–338 (2009)MathSciNetMATHGoogle Scholar
  178. 178.
    Hu, S., Papageorgiou, N.S.: Multiplicity of solutions for parametric p-Laplacian equations with nonlinearity concave near the origin. Tohoku Math. J. 62(2), 137–162 (2010)MathSciNetMATHGoogle Scholar
  179. 179.
    Iannacci, R., Nkashama, M.N.: Nonlinear boundary value problems at resonance. Nonlin. Anal. 11, 455–473 (1987)MathSciNetMATHGoogle Scholar
  180. 180.
    Iannacci, R., Nkashama, M.N.: Nonlinear two-point boundary value problems at resonance without Landesman-Lazer condition. Proc. Amer. Math. Soc. 106, 943–952 (1989)MathSciNetMATHGoogle Scholar
  181. 181.
    Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North-Holland Publishing Co., Amsterdam (1979)MATHGoogle Scholar
  182. 182.
    Ize, J., Vignoli, A.: Equivariant Degree Theory. Walter de Gruyter and Co., Berlin (2003)MATHGoogle Scholar
  183. 183.
    Jabri, Y.: The Mountain Pass Theorem: Variants, Generalizations and Some Applications. Cambridge University Press, Cambridge (2003)Google Scholar
  184. 184.
    Jebelean, P., Motreanu, D., Motreanu, V.V.: A unified approach for a class of problems involving a pseudo-monotone operator. Math. Nachr. 281, 1283–1293 (2008)MathSciNetMATHGoogle Scholar
  185. 185.
    Jiu, Q., Su, J.: Existence and multiplicity results for Dirichlet problems with p-Laplacian. J. Math. Anal. Appl. 281, 587–601 (2003)MathSciNetMATHGoogle Scholar
  186. 186.
    Jost, J.: Partial Differential Equations. Springer, New York (2002)MATHGoogle Scholar
  187. 187.
    Kačurovskiĭ, R.I.: Monotone operators and convex functionals. Uspekhi Mat. Nauk 154(94), 213–215 (1960)Google Scholar
  188. 188.
    Kačurovskiĭ, R.I.: Nonlinear monotone operators in Banach spaces. Uspekhi Mat. Nauk 232(140), 121–168 (1968)Google Scholar
  189. 189.
    Kartsatos, A.G., Skrypnik, I.V.: Topological degree theories for densely defined mappings involving operators of type (S +). Adv. Differ. Equat. 4, 413–456 (1999)MathSciNetMATHGoogle Scholar
  190. 190.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1995)MATHGoogle Scholar
  191. 191.
    Kavian, O.: Introduction à la Théorie des Points Critiques et Applications aux Problèmes Elliptiques. Springer, Paris (1993)MATHGoogle Scholar
  192. 192.
    Kenmochi, N.: Nonlinear operators of monotone type in reflexive Banach spaces and nonlinear perturbations. Hiroshima Math. J. 4, 229–263 (1974)MathSciNetMATHGoogle Scholar
  193. 193.
    Kenmochi, N.: Pseudomonotone operators and nonlinear elliptic boundary value problems. J. Math. Soc. Japan 27, 121–149 (1975)MathSciNetMATHGoogle Scholar
  194. 194.
    Khan, A.A., Motreanu, D.: Local minimizers versus X-local minimizers. Optim. Lett. 7, 1027–1033 (2013)MathSciNetMATHGoogle Scholar
  195. 195.
    Kielhöfer, H.: Bifurcation Theory: An Introduction with Applications to PDEs. Springer, New York (2004)Google Scholar
  196. 196.
    Kien, B.T., Wong, M.M., Wong, N.-C.: On the degree theory for general mappings of monotone type. J. Math. Anal. Appl. 340, 707–720 (2008)MathSciNetMATHGoogle Scholar
  197. 197.
    Kobayashi, J., Ôtani, M.: Topological degree for (S)+-mappings with maximal monotone perturbations and its applications to variational inequalities. Nonlin. Anal. 59, 147–172 (2004)MATHGoogle Scholar
  198. 198.
    Kobayashi, J., Ôtani, M.: An index formula for the degree of (S)+-mappings associated with one-dimensional p-Laplacian. Abstr. Appl. Anal. 2004, 981–995 (2004)MATHGoogle Scholar
  199. 199.
    Kobayashi, J., Ôtani, M.: Degree for subdifferential operators in Hilbert spaces. Adv. Math. Sci. Appl. 14, 307–325 (2004)MathSciNetMATHGoogle Scholar
  200. 200.
    Kobayashi, J., Ôtani, M.: The principle of symmetric criticality for non-differentiable mappings. J. Funct. Anal. 214, 428–449 (2004)MathSciNetMATHGoogle Scholar
  201. 201.
    Kondrachov, W.: Sur certaines propriétés des fonctions dans l’espace. C. R. (Doklady) Acad. Sci. URSS (N.S.) 48, 535–538 (1945)Google Scholar
  202. 202.
    Krasnosel’skiĭ, M.A.: Topological Methods in the Theory of Nonlinear Integral Equations. The Macmillan Co., New York (1964)Google Scholar
  203. 203.
    Krasnosel’skiĭ, M.A., Zabreĭko, P.P., Pustyl’nik, E.I., Sobolevskiĭ, P.E.: Integral Operators in Spaces of Summable Functions. Noordhoff International Publishing, Leiden (1976)Google Scholar
  204. 204.
    Krawcewicz, W., Marzantowicz, W.: Some remarks on the Lusternik-Schnirel’man method for nondifferentiable functionals invariant with respect to a finite group action. Rocky Mountain J. Math. 20, 1041–1049 (1990)MathSciNetMATHGoogle Scholar
  205. 205.
    Krawcewicz, W., Wu, J.: Theory of Degrees with Applications to Bifurcations and Differential Equations. Wiley Inc., New York (1997)MATHGoogle Scholar
  206. 206.
    Kufner, A.: Weighted Sobolev Spaces. Wiley Inc., New York (1985)MATHGoogle Scholar
  207. 207.
    Kufner, A., John, O., Fučík, S.: Function Spaces. Noordhoff International Publishing, Leyden (1977)MATHGoogle Scholar
  208. 208.
    Kuiper, N.H.: C 1-equivalence of functions near isolated critical points. In: Symposium on Infinite-Dimensional Topology (Louisiana State University, Baton Rouge, LA, 1967). Annals of Mathematics Studies, vol. 69, pp. 199–218. Princeton University Press, Princeton (1972)Google Scholar
  209. 209.
    Kuo, C.C.: On the solvability of a nonlinear second-order elliptic equation at resonance. Proc. Amer. Math. Soc. 124, 83–87 (1996)MathSciNetMATHGoogle Scholar
  210. 210.
    Kyritsi, S.T., Papageorgiou, N.S.: Nonsmooth critical point theory on closed convex sets and nonlinear hemivariational inequalities. Nonlin. Anal. 61, 373–403 (2005)MathSciNetMATHGoogle Scholar
  211. 211.
    Kyritsi, S.T., Papageorgiou, N.S.: Solutions for doubly resonant nonlinear non-smooth periodic problems. Proc. Edinb. Math. Soc. 48(2), 199–211 (2005)MathSciNetMATHGoogle Scholar
  212. 212.
    Kyritsi, S.T., Papageorgiou, N.S.: Positive solutions for the periodic scalar p-Laplacian: existence and uniqueness. Taiwanese J. Math. 16, 1345–1361 (2012)MathSciNetMATHGoogle Scholar
  213. 213.
    Kyritsi, S.T., Papageorgiou, N.S.: Multiple solutions for nonlinear elliptic equations with an asymmetric reaction term. Discrete Contin. Dyn. Syst. Ser. A 33, 2469–2494 (2013)MathSciNetMATHGoogle Scholar
  214. 214.
    Kyritsi, S.T., O’Regan, D., Papageorgiou, N.S.: Existence of multiple solutions for nonlinear Dirichlet problems with a nonhomogeneous differential operator. Adv. Nonlin. Stud. 10, 631–657 (2010)MathSciNetMATHGoogle Scholar
  215. 215.
    Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic, New York (1968)MATHGoogle Scholar
  216. 216.
    Landesman, E.M., Lazer, A.C.: Nonlinear perturbations of linear elliptic boundary value problems at resonance. J. Math. Mech. 19, 609–623 (1969/1970)Google Scholar
  217. 217.
    Laurent, P.-J.: Approximation et Optimisation. Hermann, Paris (1972)MATHGoogle Scholar
  218. 218.
    Lax, P.D., Milgram, A.N.: Parabolic equations. In: Contributions to the Theory of Partial Differential Equations, pp. 167–190. Princeton University Press, Princeton (1954)Google Scholar
  219. 219.
    Lazer, A.C., Solimini, S.: Nontrivial solutions of operator equations and Morse indices of critical points of min-max type. Nonlin. Anal. 12, 761–775 (1988)MathSciNetMATHGoogle Scholar
  220. 220.
    Lê, A.: Eigenvalue problems for the p-Laplacian. Nonlin. Anal. 64, 1057–1099 (2006)MATHGoogle Scholar
  221. 221.
    Lebourg, G.: Valeur moyenne pour gradient généralisé. C. R. Acad. Sci. Paris Sér. A 281, 795–797 (1975)MathSciNetMATHGoogle Scholar
  222. 222.
    Leray, J., Schauder, J.: Topologie et équations fonctionnelles. Ann. Sci. École Norm. Sup. 51(3), 45–78 (1934)MathSciNetGoogle Scholar
  223. 223.
    Li, C.: The existence of infinitely many solutions of a class of nonlinear elliptic equations with Neumann boundary condition for both resonance and oscillation problems. Nonlin. Anal. 54, 431–443 (2003)MATHGoogle Scholar
  224. 224.
    Li, S.J., Willem, M.: Applications of local linking to critical point theory. J. Math. Anal. Appl. 189, 6–32 (1995)MathSciNetMATHGoogle Scholar
  225. 225.
    Li, G., Yang, C.: The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of p-Laplacian type without the Ambrosetti-Rabinowitz condition. Nonlin. Anal. 72, 4602–4613 (2010)MATHGoogle Scholar
  226. 226.
    Li, S., Wu, S., Zhou, H.-S.: Solutions to semilinear elliptic problems with combined nonlinearities. J. Differ. Equat. 185, 200–224 (2002)MathSciNetMATHGoogle Scholar
  227. 227.
    Liapounoff, A.: Problème général de la stabilité du mouvement. Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. 9(2), 203–474 (1907)MathSciNetMATHGoogle Scholar
  228. 228.
    Lieberman, G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlin. Anal. 12, 1203–1219 (1988)MathSciNetMATHGoogle Scholar
  229. 229.
    Lindqvist, P.: On the equation \(\mathrm{div}\,(\vert \nabla {u\vert }^{p-2}\nabla u) +\lambda \vert {u\vert }^{p-2}u = 0\). Proc. Amer. Math. Soc. 109, 157–164 (1990)MathSciNetMATHGoogle Scholar
  230. 230.
    Ling, J.: Unique continuation for a class of degenerate elliptic operators. J. Math. Anal. Appl. 168, 511–517 (1992)MathSciNetMATHGoogle Scholar
  231. 231.
    Liu, J.: The Morse index of a saddle point. Syst. Sci. Math. Sci. 2, 32–39 (1989)MATHGoogle Scholar
  232. 232.
    Liu, J., Li, S.: An existence theorem for multiple critical points and its application. Kexue Tongbao (Chinese) 29, 1025–1027 (1984)MathSciNetGoogle Scholar
  233. 233.
    Liu, Z., Motreanu, D.: A class of variational-hemivariational inequalities of elliptic type. Nonlinearity 23, 1741–1752 (2010)MathSciNetMATHGoogle Scholar
  234. 234.
    Liu, J., Su, J.: Remarks on multiple nontrivial solutions for quasi-linear resonant problems. J. Math. Anal. Appl. 258 209–222 (2001)MathSciNetMATHGoogle Scholar
  235. 235.
    Liu, Z., Wang, Z.-Q.: Sign-changing solutions of nonlinear elliptic equations. Front. Math. China 3, 221–238 (2008)MathSciNetMATHGoogle Scholar
  236. 236.
    Liu, J., Wu, S.: Calculating critical groups of solutions for elliptic problem with jumping nonlinearity. Nonlin. Anal. 49,779–797 (2002)MATHGoogle Scholar
  237. 237.
    Lloyd, N.G.: Degree Theory. Cambridge University Press, Cambridge (1978)MATHGoogle Scholar
  238. 238.
    Lucia, M., Prashanth, S.: Strong comparison principle for solutions of quasilinear equations. Proc. Amer. Math. Soc. 132, 1005–1011 (2004)MathSciNetMATHGoogle Scholar
  239. 239.
    Manásevich, R., Mawhin, J.: Periodic solutions for nonlinear systems with p-Laplacian-like operators. J. Differ. Equat. 145, 367–393 (1998)MATHGoogle Scholar
  240. 240.
    Manes, A., Micheletti, A.M.: Un’estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine. Boll. Un. Mat. Ital. 7(4), 285–301 (1973)MathSciNetMATHGoogle Scholar
  241. 241.
    Marano, S.A., Motreanu, D.: Infinitely many critical points of non-differentiable functions and applications to a Neumann-type problem involving the p-Laplacian. J. Differ. Equat. 182, 108–120 (2002)MathSciNetMATHGoogle Scholar
  242. 242.
    Marano, S.A., Motreanu, D.: A deformation theorem and some critical point results for non-differentiable functions. Topol. Methods Nonlin. Anal. 22, 139–158 (2003)MathSciNetMATHGoogle Scholar
  243. 243.
    Marano, S.A., Motreanu, D.: Critical points of non-smooth functions with a weak compactness condition. J. Math. Anal. Appl. 358, 189–201 (2009)MathSciNetMATHGoogle Scholar
  244. 244.
    Marano, S.A., Papageorgiou, N.S.: Constant-sign and nodal solutions of coercive (p, q)-Laplacian problems. Nonlin. Anal. 77, 118–129 (2013)MathSciNetGoogle Scholar
  245. 245.
    Marcus, M., Mizel, V.J.: Absolute continuity on tracks and mappings of Sobolev spaces. Arch. Rational Mech. Anal. 45, 294–320 (1972)MathSciNetMATHGoogle Scholar
  246. 246.
    Marcus, M., Mizel, V.J.: Continuity of certain Nemitsky operators on Sobolev spaces and the chain rule. J. Anal. Math. 28, 303–334 (1975)MathSciNetMATHGoogle Scholar
  247. 247.
    Marcus, M., Mizel, V.J.: Every superposition operator mapping one Sobolev space into another is continuous. J. Funct. Anal. 33, 217–229 (1979)MathSciNetMATHGoogle Scholar
  248. 248.
    Margulies, C.A., Margulies, W.: An example of the Fučik spectrum. Nonlin. Anal. 29, 1373–1378 (1997)MathSciNetMATHGoogle Scholar
  249. 249.
    Marino, A., Prodi, G.: Metodi perturbativi nella teoria di Morse. Boll. Un. Mat. Ital. 11(4, suppl. 3), 1–32 (1975)Google Scholar
  250. 250.
    Martio, O.: Counterexamples for unique continuation. Manuscripta Math. 60, 21–47 (1988)MathSciNetMATHGoogle Scholar
  251. 251.
    Mawhin, J.: Semicoercive monotone variational problems. Acad. Roy. Belg. Bull. Cl. Sci. 73(5), 118–130 (1987)MathSciNetMATHGoogle Scholar
  252. 252.
    Mawhin, J.: Forced second order conservative systems with periodic nonlinearity. Ann. Inst. H. Poincaré Anal. Non Linéaire 6, 415–434 (1989)MathSciNetMATHGoogle Scholar
  253. 253.
    Mawhin, J., Willem, M.: Critical point theory and Hamiltonian systems. Springer, New York (1989)MATHGoogle Scholar
  254. 254.
    Mawhin, J., Ward, J.R., Willem, M.: Variational methods and semilinear elliptic equations. Arch. Rational Mech. Anal. 95, 269–277 (1986)MathSciNetMATHGoogle Scholar
  255. 255.
    Maz’ja, V.G.: Sobolev Spaces. Springer, Berlin (1985)MATHGoogle Scholar
  256. 256.
    Megginson, R.E.: An Introduction to Banach Space Theory. Springer, New York (1998)MATHGoogle Scholar
  257. 257.
    Meyers, N.G., Serrin, J.: H = W. Proc. Nat. Acad. Sci. USA 51, 1055–1056 (1964)MathSciNetMATHGoogle Scholar
  258. 258.
    Minty, G.J.: Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29, 341–346 (1962)MathSciNetMATHGoogle Scholar
  259. 259.
    Minty, G.J.: On a “monotonicity” method for the solution of non-linear equations in Banach spaces. Proc. Nat. Acad. Sci. USA 50, 1038–1041 (1963)MathSciNetMATHGoogle Scholar
  260. 260.
    Miyagaki, O.H., Souto, M.A.S.: Superlinear problems without Ambrosetti and Rabinowitz growth condition. J. Differ. Equat. 245, 3628–3638 (2008)MathSciNetMATHGoogle Scholar
  261. 261.
    Miyajima, S., Motreanu, D., Tanaka, M.: Multiple existence results of solutions for the Neumann problems via super- and sub-solutions. J. Funct. Anal. 262, 1921–1953 (2012)MathSciNetGoogle Scholar
  262. 262.
    Montenegro, M.: Strong maximum principles for supersolutions of quasilinear elliptic equations. Nonlin. Anal. 37, 431–448 (1999)MathSciNetMATHGoogle Scholar
  263. 263.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I: Basic Theory. Springer, Berlin (2006)Google Scholar
  264. 264.
    Moroz, V.: Solutions of superlinear at zero elliptic equations via Morse theory. Topol. Methods Nonlin. Anal. 10, 387–397 (1997)MathSciNetMATHGoogle Scholar
  265. 265.
    Morrey, C.B.: Functions of several variables and absolute continuity II. Duke Math. J. 6, 187–215 (1940)MathSciNetGoogle Scholar
  266. 266.
    Morrey, C.B.: Multiple Integrals in the Calculus of Variations. Springer, New York (1966)MATHGoogle Scholar
  267. 267.
    Morse, M.: Relations between the critical points of a real function of n independent variables. Trans. Amer. Math. Soc. 27, 345–396 (1925)MathSciNetMATHGoogle Scholar
  268. 268.
    Morse, M.: The Calculus of Variations in the Large, vol. 18, p. IX+368. American Mathematical Society Colloquium Publications, New York (1934)Google Scholar
  269. 269.
    Moser, J.: A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations. Comm. Pure Appl. Math. 13, 457–468 (1960)MathSciNetMATHGoogle Scholar
  270. 270.
    Motreanu, V.V.: Multiplicity of solutions for variable exponent Dirichlet problem with concave term. Discrete Contin. Dyn. Syst. 5, 845–855 (2012)MathSciNetMATHGoogle Scholar
  271. 271.
    Motreanu, D.: Three solutions with precise sign properties for systems of quasilinear elliptic equations. Discrete Contin. Dyn. Syst. Ser. S 5, 831–843 (2012)MathSciNetMATHGoogle Scholar
  272. 272.
    Motreanu, D., Motreanu, V.V.: Coerciveness property for a class of non-smooth functionals. Z. Anal. Anwend. 19, 1087–1093 (2000)MathSciNetMATHGoogle Scholar
  273. 273.
    Motreanu, D., Motreanu, V.V.: Nonsmooth variational problems in the limit case and duality. J. Global Optim. 29, 439–453 (2004)MathSciNetMATHGoogle Scholar
  274. 274.
    Motreanu, D., Panagiotopoulos, P.D.: Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities. Kluwer Academic Publishers, Dordrecht (1999)MATHGoogle Scholar
  275. 275.
    Motreanu, D., Papageorgiou, N.S.: Existence and multiplicity of solutions for Neumann problems. J. Differ. Equat. 232, 1–35 (2007)MathSciNetMATHGoogle Scholar
  276. 276.
    Motreanu, D., Papageorgiou, N.S.: Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator. Proc. Am. Math. Soc. 139, 3527–3535 (2011)MathSciNetMATHGoogle Scholar
  277. 277.
    Motreanu, D., Perera, K.: Multiple nontrivial solutions of Neumann p-Laplacian systems. Topol. Methods Nonlin. Anal. 34, 41–48 (2009)MathSciNetGoogle Scholar
  278. 278.
    Motreanu, D., Rădulescu, V.: Variational and Non-Variational Methods in Nonlinear Analysis and Boundary Value Problems. Kluwer Academic Publishers, Dordrecht (2003)MATHGoogle Scholar
  279. 279.
    Motreanu, D., Tanaka, M.: Sign-changing and constant-sign solutions for p-Laplacian problems with jumping nonlinearities. J. Differ. Equat. 249, 3352–3376 (2010)MathSciNetMATHGoogle Scholar
  280. 280.
    Motreanu, D., Tanaka, M.: Existence of solutions for quasilinear elliptic equations with jumping nonlinearities under the Neumann boundary condition. Calc. Var. Partial Differ. Equat. 43, 231–264 (2012)MathSciNetMATHGoogle Scholar
  281. 281.
    Motreanu, D., Tanaka, M.: Generalized eigenvalue problems of nonhomogeneous elliptic operators and their application. Pacific J. Math. 265, 151–184 (2013)MathSciNetMATHGoogle Scholar
  282. 282.
    Motreanu, D., Winkert, P.: On the Fučik spectrum for the p-Laplacian with Robin boundary condition. Nonlin. Anal. 74, 4671–4681 (2011)MathSciNetMATHGoogle Scholar
  283. 283.
    Motreanu, D., Winkert, P.: The Fučík spectrum for the negative p-Laplacian with different boundary conditions. In: Nonlinear Analysis, pp. 471–485. Springer, New York (2012)Google Scholar
  284. 284.
    Motreanu, D., Zhang, Z.: Constant sign and sign changing solutions for systems of quasilinear elliptic equations. Set-Valued Var. Anal. 19, 255–269 (2011)MathSciNetMATHGoogle Scholar
  285. 285.
    Motreanu, D., Motreanu, V.V., Paşca, D.: A version of Zhong’s coercivity result for a general class of nonsmooth functionals. Abstr. Appl. Anal. 7, 601–612 (2002)MathSciNetMATHGoogle Scholar
  286. 286.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Periodic solutions for nonautonomous systems with nonsmooth quadratic or superquadratic potential. Topol. Methods Nonlin. Anal. 24, 269–296 (2004)MathSciNetMATHGoogle Scholar
  287. 287.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Existence of solutions for strongly nonlinear elliptic differential inclusions with unilateral constraints. Adv. Differ. Equat. 10, 961–982 (2005)MathSciNetMATHGoogle Scholar
  288. 288.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Multiple nontrivial solutions for nonlinear eigenvalue problems. Proc. Amer. Math. Soc. 135, 3649–3658 (2007)MathSciNetMATHGoogle Scholar
  289. 289.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: A unified approach for multiple constant sign and nodal solutions. Adv. Differ. Equat. 12, 1363–1392 (2007)MathSciNetMATHGoogle Scholar
  290. 290.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Two nontrivial solutions for periodic systems with indefinite linear part. Discrete Contin. Dyn. Syst. 19, 197–210 (2007)MathSciNetMATHGoogle Scholar
  291. 291.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: A degree theoretic approach for multiple solutions of constant sign for nonlinear elliptic equations. Manuscripta Math. 124, 507–531 (2007)MathSciNetMATHGoogle Scholar
  292. 292.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: A multiplicity theorem for problems with the p-Laplacian. Nonlin. Anal. 68, 1016–1027 (2008)MathSciNetMATHGoogle Scholar
  293. 293.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Positive solutions and multiple solutions at non-resonance, resonance and near resonance for hemivariational inequalities with p-Laplacian. Trans. Amer. Math. Soc. 360, 2527–2545 (2008)MathSciNetMATHGoogle Scholar
  294. 294.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Nonlinear Neumann problems near resonance. Indiana Univ. Math. J. 58, 1257–1279 (2009)MathSciNetMATHGoogle Scholar
  295. 295.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Multiple solutions for Dirichlet problems which are superlinear at + and (sub-)linear at −. Commun. Appl. Anal. 13, 341–357 (2009)Google Scholar
  296. 296.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Multiple solutions for resonant nonlinear periodic equations. NoDEA - Nonlin. Differ. Equat. Appl. 17, 535–557 (2010)MathSciNetMATHGoogle Scholar
  297. 297.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Existence and multiplicity of solutions for asympotically linear, noncoercive elliptic equations. Monatsh. Math. 159, 59–80 (2010)MathSciNetMATHGoogle Scholar
  298. 298.
    Motreanu, D., Motreanu, V.V., Turinici, M.: Coerciveness property for conical nonsmooth functionals. J. Optim. Theory Appl. 145, 148–163 (2010)MathSciNetMATHGoogle Scholar
  299. 299.
    Motreanu, D., O’Regan, D., Papageorgiou, N.S.: A unified treatment using critical point methods of the existence of multiple solutions for superlinear and sublinear Neumann problems. Commun. Pure Appl. Anal. 10, 1791–1816 (2011)MathSciNetMATHGoogle Scholar
  300. 300.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Nonautonomous resonant periodic systems with indefinite linear part and a nonsmooth potential. Commun. Pure Appl. Anal. 10, 1401–1414 (2011)MathSciNetMATHGoogle Scholar
  301. 301.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Multiple constant sign and nodal solutions for nonlinear Neumann eigenvalue problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10(5), 729–755 (2011)MathSciNetMATHGoogle Scholar
  302. 302.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: On p-Laplace equations with concave terms and asymmetric perturbations. Proc. Roy. Soc. Edinb. Sect. A 141, 171–192 (2011)MathSciNetMATHGoogle Scholar
  303. 303.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: On resonant Neumann problems. Math. Ann. 354, 1117–1145 (2012)MathSciNetMATHGoogle Scholar
  304. 304.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Existence and nonexistence of positive solutions for parametric Neumann problems with p-Laplacian. Tohoku Math. J. (to appear)Google Scholar
  305. 305.
    Mugnai, D., Papageorgiou, N.S.: Resonant nonlinear Neumann problems with indefinite weight. Ann. Sc. Norm. Super. Pisa Cl. Sci. 11(5), 729–788 (2012)MathSciNetMATHGoogle Scholar
  306. 306.
    Nagumo, M.: A theory of degree of mapping based on infinitesimal analysis. Amer. J. Math. 73, 485–496 (1951)MathSciNetMATHGoogle Scholar
  307. 307.
    Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Dekker Inc., New York (1995)Google Scholar
  308. 308.
    Ni, W.-M., Wang, X.: On the first positive Neumann eigenvalue. Discrete Contin. Dyn. Syst. 17, 1–19 (2007)MathSciNetMATHGoogle Scholar
  309. 309.
    Nirenberg, L.: Topics in Nonlinear Functional Analysis. New York University Courant Institute of Mathematical Sciences, New York (2001)MATHGoogle Scholar
  310. 310.
    Njoku, F.I., Zanolin, F.: Positive solutions for two-point BVPs: existence and multiplicity results. Nonlin. Anal. 13, 1329–1338 (1989)MathSciNetMATHGoogle Scholar
  311. 311.
    Nussbaum, R.D.: The fixed point index for local condensing maps. Ann. Mat. Pura Appl. 89(4), 217–258 (1971)MathSciNetMATHGoogle Scholar
  312. 312.
    Ôtani, M.: Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations. J. Funct. Anal. 76, 140–159 (1988)MathSciNetMATHGoogle Scholar
  313. 313.
    Palais, R.S.: Morse theory on Hilbert manifolds. Topology 2, 299–340 (1963)MathSciNetMATHGoogle Scholar
  314. 314.
    Palais, R.S.: Lusternik-Schnirelman theory on Banach manifolds. Topology 5, 115–132 (1966)MathSciNetMATHGoogle Scholar
  315. 315.
    Palais, R.S.: The principle of symmetric criticality. Comm. Math. Phys. 69, 19–30 (1979)MathSciNetMATHGoogle Scholar
  316. 316.
    Palais, R.S., Smale, S.: A generalized Morse theory. Bull. Amer. Math. Soc. 70, 165–172 (1964)MathSciNetMATHGoogle Scholar
  317. 317.
    Panagiotopoulos, P.D.: Hemivariational Inequalities. Applications in Mechanics and Engineering. Springer, Berlin (1993)MATHGoogle Scholar
  318. 318.
    Papageorgiou, N.S., Kyritsi-Yiallourou, S.T.: Handbook of Applied Analysis. Springer, New York (2009)MATHGoogle Scholar
  319. 319.
    Papageorgiou, E.H., Papageorgiou, N.S.: A multiplicity theorem for problems with the p-Laplacian. J. Funct. Anal. 244, 63–77 (2007)MathSciNetMATHGoogle Scholar
  320. 320.
    Papageorgiou, N.S., Papalini, F.: On the existence of three nontrivial solutions for periodic problems driven by the scalar p-Laplacian. Adv. Nonlin. Stud. 11, 455–471 (2011)MathSciNetMATHGoogle Scholar
  321. 321.
    Papageorgiou, N.S., Papalini, F.: Multiple solutions for nonlinear periodic systems with combined nonlinearities and a nonsmooth potential. J. Nonlin. Convex Anal. 13, 681–693 (2012)MathSciNetMATHGoogle Scholar
  322. 322.
    Papageorgiou, N.S., Staicu, V.: Multiple nontrivial solutions for doubly resonant periodic problems. Canad. Math. Bull. 53, 347–359 (2010)MathSciNetMATHGoogle Scholar
  323. 323.
    Papageorgiou, N.S., Rocha, E.M., Staicu, V.: A multiplicity theorem for hemivariational inequalities with a p-Laplacian-like differential operator. Nonlin. Anal. 69, 1150–1163 (2008)MathSciNetMATHGoogle Scholar
  324. 324.
    Pascali, D., Sburlan, S.: Nonlinear Mappings of Monotone Type. Martinus Nijhoff Publishers, The Hague (1978)MATHGoogle Scholar
  325. 325.
    Pauli, W.: Theory of Relativity. Pergamon, New York (1958)MATHGoogle Scholar
  326. 326.
    Perera, K.: Homological local linking. Abstr. Appl. Anal. 3, 181–189 (1998)MathSciNetMATHGoogle Scholar
  327. 327.
    Perera, K.: Nontrivial critical groups in p-Laplacian problems via the Yang index. Topol. Methods Nonlin. Anal. 21, 301–309 (2003)MathSciNetMATHGoogle Scholar
  328. 328.
    Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability. Lecture Notes in Mathematics, vol. 1364. Springer, Berlin (1993)Google Scholar
  329. 329.
    Pigola, S., Rigoli, M., Setti, A.G.: Maximum principles on Riemannian manifolds and applications. Mem. Amer. Math. Soc. 174, 822 (2005)MathSciNetGoogle Scholar
  330. 330.
    Pucci, P., Serrin, J.: A mountain pass theorem. J. Differ. Equat. 60, 142–149 (1985)MathSciNetMATHGoogle Scholar
  331. 331.
    Pucci, P., Serrin, J.: The structure of the critical set in the mountain pass theorem. Trans. Amer. Math. Soc. 299, 115–132 (1987)MathSciNetMATHGoogle Scholar
  332. 332.
    Pucci, P., Serrin, J.: The Maximum Principle. Birkhäuser, Basel (2007)MATHGoogle Scholar
  333. 333.
    Qian, A.: Existence of infinitely many nodal solutions for a superlinear Neumann boundary value problem. Bound. Value Probl. 2005, 329–335 (2005)MATHGoogle Scholar
  334. 334.
    Rabinowitz, P.H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7, 487–513 (1971)MathSciNetMATHGoogle Scholar
  335. 335.
    Rabinowitz, P.H.: A note on topological degree for potential operators. J. Math. Anal. Appl. 51, 483–492 (1975)MathSciNetMATHGoogle Scholar
  336. 336.
    Rabinowitz, P.H.: Some minimax theorems and applications to nonlinear partial differential equations. In: Nonlinear Analysis (collection of papers in honor of Erich H. Rothe), pp. 161–177. Academic, New York (1978)Google Scholar
  337. 337.
    Rabinowitz, P.H.: Periodic solutions of Hamiltonian systems. Comm. Pure Appl. Math. 31, 157–184 (1978)MathSciNetGoogle Scholar
  338. 338.
    Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. American Mathematical Society, Washington (1986)Google Scholar
  339. 339.
    Rellich, F.: Ein Satz über mittlere Konvergenz. Nachrichten Göttingen 1930, 30–35 (1930)MATHGoogle Scholar
  340. 340.
    Ricceri, B.: Infinitely many solutions of the Neumann problem for elliptic equations involving the p-Laplacian. Bull. London Math. Soc. 33, 331–340 (2001)MathSciNetMATHGoogle Scholar
  341. 341.
    Riesz, F.: Über lineare Funktionalgleichungen. Acta Math. 41, 71–98 (1916)MathSciNetMATHGoogle Scholar
  342. 342.
    Roberts, A.W., Varberg, D.E.: Convex Functions. Academic, New York (1973)MATHGoogle Scholar
  343. 343.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)MATHGoogle Scholar
  344. 344.
    Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Amer. Math. Soc. 149, 75–88 (1970)MathSciNetMATHGoogle Scholar
  345. 345.
    Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pacific J. Math. 33, 209–216 (1970)MathSciNetMATHGoogle Scholar
  346. 346.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)Google Scholar
  347. 347.
    Roselli, P., Sciunzi, B.: A strong comparison principle for the p-Laplacian. Proc. Amer. Math. Soc. 135, 3217–3224 (2007)MathSciNetMATHGoogle Scholar
  348. 348.
    Rothe, E.H.: Morse theory in Hilbert space. Rocky Mountain J. Math. 3, 251–274 (1973)MathSciNetMATHGoogle Scholar
  349. 349.
    Rynne, B.P.: Spectral properties of p-Laplacian problems with Neumann and mixed-type multi-point boundary conditions. Nonlin. Anal. 74, 1471–1484 (2011)MathSciNetMATHGoogle Scholar
  350. 350.
    Schauder, J.: Der Fixpunktsatz in Funktionalräumen. Studia 2, 171–180 (1930)MATHGoogle Scholar
  351. 351.
    Schechter, M.: The Fučík spectrum. Indiana Univ. Math. J. 43, 1139–1157 (1994)MathSciNetMATHGoogle Scholar
  352. 352.
    Schechter, M.: Infinite-dimensional linking. Duke Math. J. 94, 573–595 (1998)MathSciNetMATHGoogle Scholar
  353. 353.
    Schechter, M.: Linking Methods in Critical Point Theory. Birkhäuser, Boston (1999)MATHGoogle Scholar
  354. 354.
    Schechter, M.: Principles of Functional Analysis. American Mathematical Society, Providence (2002)Google Scholar
  355. 355.
    Schechter, M., Zou, W.: Superlinear problems. Pacific J. Math. 214, 145–160 (2004)MathSciNetMATHGoogle Scholar
  356. 356.
    Schmidt, E.: Zur Theorie der linearen und nichtlinearen Integralgleichungen III, Teil: Über die Auflösung der nichtlinearen Integralgleichungen und die Verzweigung ihrer Lösungen. Math. Ann. 65, 370–399 (1908)MATHGoogle Scholar
  357. 357.
    Schwartz, L.: Théorie des Distributions: Tome I. Hermann and Cie., Paris (1950)MATHGoogle Scholar
  358. 358.
    Schwartz, L.: Théorie des Distributions: Tome II. Hermann and Cie., Paris (1951)MATHGoogle Scholar
  359. 359.
    Serrin, J.: Local behavior of solutions of quasi-linear equations. Acta Math. 111, 247–302 (1964)MathSciNetMATHGoogle Scholar
  360. 360.
    Skrypnik, I.V.: Nonlinear Elliptic Boundary Value Problems. Teubner Verlagsgesellschaft, Leipzig (1986)MATHGoogle Scholar
  361. 361.
    Silva, E.A., Teixeira, M.A.: A version of Rolle’s theorem and applications. Bol. Soc. Brasil. Mat. (N.S.) 29, 301–327 (1998)Google Scholar
  362. 362.
    Smale, S.: Morse theory and a non-linear generalization of the Dirichlet problem. Ann. of Math. 80(2), 382–396 (1964)MathSciNetMATHGoogle Scholar
  363. 363.
    Sobolev, S.L.: Applications of Functional Analysis in Mathematical Physics. Amer. Math. Soc., Providence (1963)MATHGoogle Scholar
  364. 364.
    Sobolev, S.L.: On a theorem of functional analysis. Amer. Math. Soc. Transl. 34, 39–68 (1963)MathSciNetMATHGoogle Scholar
  365. 365.
    Spanier, E.H.: Algebraic Topology. McGraw-Hill Book Co., New York (1966)MATHGoogle Scholar
  366. 366.
    Stampacchia, G.: Équations Elliptiques du Second Ordre à Coefficients Discontinus. Les Presses de l’Université de Montréal, Montreal (1966)MATHGoogle Scholar
  367. 367.
    Struwe, M.: Variational Methods. Springer, Berlin (1996)MATHGoogle Scholar
  368. 368.
    Su, J., Zhao, L.: An elliptic resonance problem with multiple solutions. J. Math. Anal. Appl. 319, 604–616 (2006)MathSciNetMATHGoogle Scholar
  369. 369.
    Szulkin, A.: Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 3, 77–109 (1986)MathSciNetMATHGoogle Scholar
  370. 370.
    Tang, C.-L.: Periodic solutions for nonautonomous second order systems with sublinear nonlinearity. Proc. Amer. Math. Soc. 126, 3263–3270 (1998)MathSciNetMATHGoogle Scholar
  371. 371.
    Tang, C.-L., Wu, X.-P.: Periodic solutions for second order systems with not uniformly coercive potential. J. Math. Anal. Appl. 259, 386–397 (2001)MathSciNetGoogle Scholar
  372. 372.
    Tang, C.-L., Wu, X.-P.: Periodic solutions for a class of nonautonomous subquadratic second order Hamiltonian systems. J. Math. Anal. Appl. 275, 870–882 (2002)MathSciNetMATHGoogle Scholar
  373. 373.
    Tang, C.-L., Wu, X.-P.: Existence and multiplicity for solutions of Neumann problem for semilinear elliptic equations. J. Math. Anal. Appl. 288, 660–670 (2003)MathSciNetMATHGoogle Scholar
  374. 374.
    Tartar, L.: An Introduction to Sobolev Spaces and Interpolation Spaces. Springer, Berlin (2007)MATHGoogle Scholar
  375. 375.
    Thews, K.: Nontrivial solutions of elliptic equations at resonance. Proc. Roy. Soc. Edinb. Sect. A 85, 119–129 (1980)MathSciNetMATHGoogle Scholar
  376. 376.
    Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equat. 51, 126–150 (1984)MathSciNetMATHGoogle Scholar
  377. 377.
    Troyanski, S.L.: On locally uniformly convex and differentiable norms in certain non-separable Banach spaces. Studia Math. 37, 173–180 (1971)MathSciNetMATHGoogle Scholar
  378. 378.
    Trudinger, N.S.: On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math. 20, 721–747 (1967)MathSciNetMATHGoogle Scholar
  379. 379.
    Vázquez, J. L.: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12, 191–202 (1984)MathSciNetMATHGoogle Scholar
  380. 380.
    Wang, J.: The existence of positive solutions for the one-dimensional p-Laplacian. Proc. Amer. Math. Soc. 125, 2275–2283 (1997)MathSciNetMATHGoogle Scholar
  381. 381.
    Weyl, H.: Space, Time, Matter. Dover Publications, New York (1951)MATHGoogle Scholar
  382. 382.
    Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)MATHGoogle Scholar
  383. 383.
    Wu, X., Tan, K.-K.: On existence and multiplicity of solutions of Neumann boundary value problems for quasi-linear elliptic equations. Nonlin. Anal. 65, 1334–1347 (2006)MathSciNetMATHGoogle Scholar
  384. 384.
    Yang, C.-T.: On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobô and Dyson. I. Ann. Math. 60(2), 262–282 (1954)MATHGoogle Scholar
  385. 385.
    Yang, X.: Multiple periodic solutions of a class of p-Laplacian. J. Math. Anal. Appl. 314, 17–29 (2006)MathSciNetMATHGoogle Scholar
  386. 386.
    Zeidler, E.: The Ljusternik-Schnirelman theory for indefinite and not necessarily odd nonlinear operators and its applications. Nonlin. Anal. 4, 451–489 (1980)MathSciNetMATHGoogle Scholar
  387. 387.
    Zeidler, E.: Nonlinear Functional Analysis and Its Applications III: Variational Methods and Optimization. Springer, New York (1985)MATHGoogle Scholar
  388. 388.
    Zeidler, E.: Nonlinear Functional Analysis and Its Applications II/A: Linear Monotone Operators. Springer, New York (1990)MATHGoogle Scholar
  389. 389.
    Zeidler, E.: Nonlinear Functional Analysis and Its Applications II/B: Nonlinear Monotone Operators. Springer-Verlag, New York (1990)MATHGoogle Scholar
  390. 390.
    Zhang, M.: The rotation number approach to eigenvalues of the one-dimensional p-Laplacian with periodic potentials. J. London Math. Soc. 64(2), 125–143 (2001)MathSciNetMATHGoogle Scholar
  391. 391.
    Zhang, Q.: A strong maximum principle for differential equations with nonstandard p(x)-growth conditions. J. Math. Anal. Appl. 312, 24–32 (2005)MathSciNetMATHGoogle Scholar
  392. 392.
    Zhang, Z., Li, S.: On sign-changing and multiple solutions of the p-Laplacian. J. Funct. Anal. 197, 447–468 (2003)MathSciNetMATHGoogle Scholar
  393. 393.
    Zhang, Q., Li, G.: On a class of second order differential inclusions driven by the scalar p-Laplacian. Nonlin. Anal. 72, 151–163 (2010)MATHGoogle Scholar
  394. 394.
    Zhang, Z., Chen, J., Li, S.: Construction of pseudo-gradient vector field and sign-changing multiple solutions involving p-Laplacian. J. Differ. Equat. 201, 287–303 (2004)MathSciNetMATHGoogle Scholar
  395. 395.
    Zhong, C.-K.: A generalization of Ekeland’s variational principle and application to the study of the relation between the weak P.S. condition and coercivity. Nonlin. Anal. 29, 1421–1431 (1997)Google Scholar
  396. 396.
    Ziemer, W.P.: Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation. Springer, New York (1989)MATHGoogle Scholar
  397. 397.
    Zou, W.: Variant fountain theorems and their applications. Manuscripta Math. 104, 343–358 (2001)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2014

Authors and Affiliations

  • Dumitru Motreanu
    • 1
  • Viorica Venera Motreanu
    • 2
  • Nikolaos Papageorgiou
    • 3
  1. 1.Department of MathematicsUniversity of PerpignanPerpignanFrance
  2. 2.Department of MathematicsBen-Gurion University of the NegevBeer-ShevaIsrael
  3. 3.Department of MathematicsNational Technical UniversityAthensGreece

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