Nonlinear flux “concave–convex” problems: a fibering method approach


This paper studies the nonlinear flux problem:

where \(\varDelta _p\) stands for the p-Laplacian operator, \(\varOmega \subset {\mathbb {R}}^N\) is a bounded smooth domain, \(\lambda\) is a positive parameter and \(\nu\) stands for the outer unit normal at \(\partial \varOmega\). The exponents qr are assumed to vary in the concave convex regime \(1< q< p < r\) while \(1< p < N\) and r is subcritical \(r < p^*\). Our objective here is showing the existence, for every \(0< \lambda < {{\bar{\lambda }}}\), of two different sets of infinitely many solutions of (P). The energy functional associated to the problem exhibits a different sign on each of these sets. The analysis of positive energy solutions involves the so-called fibering method (Drábek and Pohozaev in Proc R Soc Edinb Sect A 127(4):703–726, 1997). Our results have been inspired by similar ones in García-Azorero et al. (J Differ Equ 198(1):91–128, 2004), García-Azorero and Peral (Trans Am Math Soc 323(2):877–895, 1991) and El Hamidi (Commun Pure Appl Anal 3(2):253–265, 2004). This work can be considered as a natural continuation of Sabina de Lis (Differ Equ Appl 3(4):469–486, 2011), Sabina de Lis and Segura de León (Adv Nonlinear Stud 15(1):61–90, 2015) and Sabina de Lis and Segura de León (Nonlinear Anal 113:283–297, 2015). The main achievement of the latter of these works consisted in showing a global existence result of positive solutions to (P).

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  1. 1.

    Amann, H.: Lusternik–Schnirelman theory and non-linear eigenvalue problems. Math. Ann. 199, 55–72 (1972)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Ambrosetti, A., Brezis, H., Cerami, G.: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122(2), 519–543 (1994)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Ambrosetti, A., García-Azorero, J., Peral, I.: Multiplicity results for some nonlinear elliptic equations. J. Funct. Anal. 137(1), 219–242 (1996)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Boccardo, L., Escobedo, M., Peral, I.: A Dirichlet problem involving critical exponents. Nonlinear Anal. 24(11), 1639–1648 (1995)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Brown, K.J., Zhang, Y.: The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function. J. Differ. Equ. 193(2), 481–499 (2003)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Cuesta, M., Leadi, L.: On abstract indefinite concave–convex problems and applications to quasilinear elliptic equations. NoDEA Nonlinear Differ. Equ. Appl. 24(2), 31 (2017). (Art. 20)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Drábek, P., Pohozaev, S.I.: Positive solutions for the \(p\)-Laplacian: application of the fibering method. Proc. R. Soc. Edinb. Sect. A 127(4), 703–726 (1997)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    El Hamidi, A.: Multiple solutions with changing sign energy to a nonlinear elliptic equation. Commun. Pure Appl. Anal. 3(2), 253–265 (2004)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    García-Azorero, J., Peral, I.: Existence and nonuniqueness for the \(p\)-Laplacian: nonlinear eigenvalues. Commun. Partial Differ. Equ. 12(12), 1389–1430 (1987)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    García-Azorero, J., Peral, I.: Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term. Trans. Am. Math. Soc. 323(2), 877–895 (1991)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    García-Azorero, J., Peral, I.: Some results about the existence of a second positive solution in a quasilinear critical problem. Indiana Univ. Math. J. 43(3), 941–957 (1994)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    García-Azorero, J., Manfredi, J.J., Peral, I.: Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations. Commun. Contemp. Math. 2(3), 385–404 (2000)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    García-Azorero, J., Peral, I., Rossi, J.D.: A convex–concave problem with a nonlinear boundary condition. J. Differ. Equ. 198(1), 91–128 (2004)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    García-Melián, J., Rossi, J.D., Sabina de Lis, J.: A convex–concave elliptic problem with a parameter on the boundary condition. Discrete Contin. Dyn. Syst. 32(4), 1095–1124 (2012)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Krasnosel’skii, M.A.: Topological methods in the theory of nonlinear integral equations. The Macmillan Co., New York (1964)

    Google Scholar 

  16. 16.

    Lieberman, G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12(11), 1203–1219 (1988)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Lions, P.-L.: On the existence of positive solutions of semilinear elliptic equations. SIAM Rev. 24(4), 441–467 (1982)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. CBMS regional conference series in mathematics, vol. 65. American Mathematical Society, Providence (1986)

    Google Scholar 

  19. 19.

    Sabina de Lis, J.C.: A concave-convex quasilinear elliptic problem subject to a nonlinear boundary condition. Differ. Equ. Appl. 3(4), 469–486 (2011)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Sabina de Lis, J.C., Segura de León, S.: Multiplicity of solutions to a concave–convex problem. Adv. Nonlinear Stud. 15(1), 61–90 (2015)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Sabina de Lis, J.C., Segura de León, S.: Multiplicity of solutions to a nonlinear boundary value problem of concave–convex type. Nonlinear Anal. 113, 283–297 (2015)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Struwe, M.: Variational methods. Modern surveys in mathematics, 4th edn. Springer, Berlin (2008)

    Google Scholar 

  23. 23.

    Szulkin, A.: Ljusternik–Schnirelmann theory on \({ C}^1\)-manifolds. Ann. Inst. H. Poincaré Anal. Non Linéaire 5(2), 119–139 (1988)

    MathSciNet  MATH  Google Scholar 

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The author is very grateful to the anonymous referee for his/her careful revision of the original manuscript. Supported by Dirección General de Investigación under Grant MTM2014-52822-P.

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Correspondence to José C. Sabina de Lis.

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Communicated by Julio Rossi.

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Sabina de Lis, J.C. Nonlinear flux “concave–convex” problems: a fibering method approach. Adv. Oper. Theory (2020).

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  • Variational methods
  • Minimax methods
  • Degenerate diffusion

Mathematics Subject Classification

  • 35J20
  • 35J70