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Elliptic Systems with Some Superlinear Assumption Only Around the Origin

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Abstract

In this paper, using a priori bound techniques we study existence of positive solutions of the elliptic system:

$$\begin{aligned} \left\{ \begin{array}{lll} -\text{ div }(|x|^{\alpha _1}\nabla u) = |x|^{\beta _1} f(|x|,u,v) \ \ x \in B, \\ -\text{ div }(|x|^{\alpha _2}\nabla v) = |x|^{\beta _2} g(|x|,u,v) \ \ x \in B, \\ u(x) = 0 =v(x), \ \ \ x \in \partial B. \end{array} \right. \end{aligned}$$

where B is the unitary ball centered at the origin. Assuming that f, g are nonnegative nonlinearities and that \(f(|x|,u,v)+g(|x|,u,v)\) is superlinear at 0 and at \(\infty \), we establish some results of existence of one positive solution. As an application, we establish two positive solutions for some non-homogeneous elliptic system. The main novelties here are that the nonlinearities could have growth above the critical hyperbola on some part of the domain as well as only local superlinear hypotheses at \(\infty \)

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References

  1. Boccardo, L., de Figueiredo, D.G.: Some remarks on a system of quasilinear elliptic equations. Nonlinear Differ. Equ. Appl. 9(3), 309–323 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Clément, Ph, de Figueiredo, D.G., Mitidieri, E.: Positive solutions of semilinear elliptic systems. Commun. Partial Differ. Equ. 17, 923–940 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. de Figueiredo, D.G.: Positive solutions of semilinear elliptic equations. Springer Lect. Notes Math. 957, 34–87 (1982)

    Article  Google Scholar 

  4. de Figueiredo, D.G., Felmer, P.L.: On superquadratic elliptic systems. Trans. Am. Math. Soc. 343, 99–116 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. de Figueiredo, D.G., Magalhaes, C.A.: On nonquadratic hamiltonian elliptic systems. Adv. Differ. Equ. 1(5), 881–898 (1996)

    MathSciNet  MATH  Google Scholar 

  6. de Figueiredo, D.G., do Ó, J.M., Ruf, B.: Non variational elliptic systems in dimension two: a priori bounds and existence of positive solutions. J. Fixed Point Theory Appl. 4, 77–96 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ruf, B.: Superlinear elliptic equations and systems. In: Chipot, M. (ed.) Handbook of Differential Equations and Systems, vol. 5. Elsevier, Amsterdam (2008)

    Google Scholar 

  8. Felmer, P., Manásevich, R.F., de Thélin, F.: Existence and uniqueness of positive solutions for certain quasilinear elliptic systems. Commun. Partial Differ. Equ. 17, 2013–2029 (1992)

    MathSciNet  MATH  Google Scholar 

  9. de Figueiredo, D.G.: Semilinear elliptic systems: a survey of superlinear problems. Resenhas 2(4), 373–391 (1996)

    MathSciNet  MATH  Google Scholar 

  10. Bonheure, D., Moreira dos Santos, E., Tavares, H.: Hamiltonian elliptic systems: a guide to variational frameworks. Port. Math. 71(3–4), 301–395 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. de Figueiredo, D.G., Felmer, P.: On superquadratic elliptic systems. Trans. Am. Math. Soc. 343, 99–116 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hulshof, J., van der Vorst, R.: Differential systems with strongly indefinite variational structure. J. Funct. Anal. 114, 32–58 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. Clément, Ph, Felmer, P., Mitidieri, E.: Homoclinic orbits for a class of infinite-dimensional Hamiltonian systems. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 24(2), 367–393 (1997)

    MathSciNet  MATH  Google Scholar 

  14. Clément, Ph, Van der Vorst, R.C.A.M.: On a semilinear elliptic system. Differ. Integral Equ. 8(6), 1317–1329 (1995)

    MathSciNet  MATH  Google Scholar 

  15. Ramos, M., Tavares, H.: Solutions with multiple spike patterns for an elliptic system. Calc. Var. Partial Differ. Equ. 31(1), 1–25 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Azizieh, C., Clément, P., Mitidieri, E.: Existence and a priori estimates for positive solutions of p-Laplace systems. J. Differ. Equ. 184(2), 422–442 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Chen, W., Dupaigne, L., Ghergu, M.: A new critical curve for the Lane–Emden system. Discrete Contin. Dyn. Syst. 34(6), 2469–2479 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Chen, Z., Lin, C.-S., Zou, W.: Monotonicity and nonexistence results to cooperative systems in the half space. J. Funct. Anal. 266(2), 1088–1105 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cowan, C.: Liouville theorems for stable Lane–Emden systems with biharmonic problems. Nonlinearity 26(8), 2357–2371 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Fazly, M., Ghoussoub, N.: On the Hénon–Lane–Emden conjecture. Discrete Contin. Dyn. Syst. 34(6), 2513–2533 (2014)

    MathSciNet  MATH  Google Scholar 

  21. Ruiz, D.: A priori estimates and existence of positive solutions for strongly nonlinear problems. J. Differ. Equ. 199, 96–114 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Kazdan, J.L., Kramer, R.J.: Invariant criteria for existence of solutions to second-order quasilinear elliptic equations. Commun. Pure Appl. Math. 31, 619–645 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  23. Quittner, P., Souplet, Ph: A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces. Arch. Ration. Mech. Anal. 174, 49–81 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. de Figueiredo, D., Lions, P.L., Nussbaum, R.D.: A priori estimates and existence of positive solutions of semilinear elliptic equations. J. Math. Pures Appl. 61, 41–63 (1982)

    MathSciNet  MATH  Google Scholar 

  25. Krasnoselskii, M.A.: Fixed point of cone-compressing or cone-extending operators. Sov. Math. Dokl. 1, 1285–1288 (1960)

    MathSciNet  Google Scholar 

  26. Wang, Z.-Q., Willem, M.: Caffarelli–Kohn–Nirenberg inequalities with reminder terms. J. Funct. Anal. 203, 550–568 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Xuan, B.: The solvability of Brezis–Nirenberg type problems of singular quasilinear elliptic equation. arXiv: math/0403549v1 (2004)

  28. Vasquez, M.A.: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12, 191–202 (1984)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemática y C. C., Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile

    Patricio Cerda & Pedro Ubilla

  2. Departamento de Matemática, Universidade Federal de Campina Grande, Cx Postal 10044, Campina Grande, PB, CEP 58.429-900, Brazil

    Marco Aurelio Souto

Authors
  1. Patricio Cerda
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  2. Marco Aurelio Souto
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  3. Pedro Ubilla
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Corresponding author

Correspondence to Marco Aurelio Souto.

Additional information

Communicated by Nader Masmoudi.

The first author was supported by PAI-CONICYT Grant 79140015. The second author was partially supported by CNPq—Proc. 306.082/2017-9. The third author was partially supported by FONDECYT Grants 1120524 and 1161635.

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Cerda, P., Souto, M.A. & Ubilla, P. Elliptic Systems with Some Superlinear Assumption Only Around the Origin. Ann. Henri Poincaré 19, 3031–3051 (2018). https://doi.org/10.1007/s00023-018-0714-2

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  • DOI: https://doi.org/10.1007/s00023-018-0714-2

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