Singular Neumann (pq)-equations


We consider a nonlinear parametric Neumann problem driven by the sum of a p-Laplacian and of a q-Laplacian and exhibiting in the reaction the competing effects of a singular term and of a resonant term. Using variational methods together with suitable truncation and comparison techniques, we show that for small values of the parameter the problem has at least two positive smooth solutions.

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  1. 1.

    Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints. Mem. Am. Math. Soc. 196(915), 1–70 (2008)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Cherfils, L., Il’yasov, Y.: On the stationary solutions of generalized reaction diffusion equations with \(p\)&\(q\)-Laplacian. Commun. Pure Appl. Anal. 4(1), 9–22 (2005)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Gasiński, L., Papageorgiou, N.S.: Nonlinear Analysis, Series Mathematical Analysis and Applications, vol. 9. CRC Press, Boca Raton (2006)

    MATH  Google Scholar 

  5. 5.

    Gasiński, L., Papageorgiou, N.S.: Nonlinear elliptic equations with singular terms and combined nonlinearities. Ann. Henri Poincaré 13(3), 481–512 (2012)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Giacomoni, J., Schindler, I., Takáč, P.: Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 6(1), 117–158 (2007)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Hirano, N., Saccon, C., Shioji, N.: Brezis–Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem. J. Differ. Equ. 245(8), 1997–2037 (2008)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Kyritsi, S., Papageorgiou, N.S.: Pairs of positive solutions for singular \(p\)-Laplacian equations with a \(p\)-superlinear potential. Nonlinear Anal. 73(5), 1136–1142 (2010)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Lair, A.V., Shaker, A.W.: Entire solution of a singular semilinear elliptic problem. J. Math. Anal. Appl. 200(2), 498–505 (1996)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Lieberman, G.M.: The natural generalization of the natural conditions of Ladyzhenskaya and Ural\(^\prime \)tseva for elliptic equations. Commun. Partial Differ. Equ. 16(2–3), 311–361 (1991)

    Article  Google Scholar 

  11. 11.

    Marano, S.A., Mosconi, S.J.N.: Some recent results on the Dirichlet problem for (\(p, q\))-Laplace equations. Discrete Contin. Dyn. Syst. Ser. S 11(2), 279–291 (2018)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Marano, S.A., Papageorgiou, N.S.: Positive solutions to a Dirichlet problem with \(p\)-Laplacian and concave-convex nonlinearity depending on a parameter. Commun. Pure Appl. Anal. 12(2), 815–829 (2013)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014)

    Book  Google Scholar 

  14. 14.

    Papageorgiou, N.S., Rădulescu, V.D.: Combined effects of singular and sublinear nonlinearities in some elliptic problems. Nonlinear Anal. 109, 236–244 (2014)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Papageorgiou, N.S., Rădulescu, V.D.: Multiple solutions with precise sign for nonlinear parametric Robin problems. J. Differ. Equ. 256(7), 2449–2479 (2014)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Papageorgiou, N.S., Rădulescu, V.D.: Nonlinear nonhomogeneous Robin problems with superlinear reaction term. Adv. Nonlinear Stud. 16(4), 737–764 (2016)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Papageorgiou, N.S., Rădulescu, V.D., Repovs̆, D.D.: Pairs of positive solutions for resonant singular equations with the \(p\)-Laplacian. Electron. J. Differ. Equ. 2017(249), 1–22 (2017)

    MathSciNet  Google Scholar 

  18. 18.

    Papageorgiou, N.S., Smyrlis, G.: A bifurcation-type theorem for singular nonlinear elliptic equations. Methods Appl. Anal. 22(2), 147–170 (2015)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Papageorgiou, N.S., Vetro, C.: Superlinear \((p(z), q(z))\)-equations. Complex Var. Elliptic Equ. 64(1), 8–25 (2019)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Perera, K., Zhang, Z.: Multiple positive solutions of singular \(p\)-Laplacian problems by variational methods. Bound. Value Probl. 2005(3), 377–382 (2005)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Pucci, P., Serrin, J.: The Maximum Principle. Birkhäuser, Basel (2007)

    Book  Google Scholar 

  22. 22.

    Sun, Y., Wu, S., Long, Y.: Combined effects of singular and superlinear nonlinearities in some singular boundary value problems. J. Differ. Equ. 176(2), 511–531 (2001)

    MathSciNet  Article  Google Scholar 

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The authors wish to thank a knowledgeable referee for his/her corrections and remarks.

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Correspondence to Francesca Vetro.

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Papageorgiou, N.S., Vetro, C. & Vetro, F. Singular Neumann (pq)-equations. Positivity 24, 1017–1040 (2020).

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  • Singular term
  • Resonant nonlinearity
  • Nonlinear regularity
  • Truncation and comparison
  • Nonlinear strong maximum principle
  • (p, q)-equation

Mathematics Subject Classification

  • 35J92
  • 35P30