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Asymptotic behavior of ground state radial solutions for problems involving the \(\Phi \)-Laplacian

  • Abdelwaheb DhifliEmail author
  • Rym Chemmam
  • Syrine Masmoudi


We are concerned with the existence of positive solutions to the following boundary value problem in \((0,\infty ),\)
$$\begin{aligned} \frac{1}{A}\left( A\phi \left( \left| u^{\prime }\right| \right) u^{\prime }\right) ^{\prime }=-a(t)u^{\alpha },t>0,\left( A\phi \left( \left| u^{\prime }\right| \right) u^{\prime }\right) \left( 0\right) =0\text { and}\lim \nolimits _{t\rightarrow +\infty }u(t)=0, \end{aligned}$$
where \(\alpha \ge 0,\)\(\phi \) is a nonnegative continuously differentiable function on \(\left[ 0,\infty \right) \), A is a continuous function on \( \left[ 0,\infty \right) \), differentiable, positive on \(\left( 0,\infty \right) \) and a is a nonnegative function satisfying some appropriate assumptions related to Karamata regular variation theory. We give also, estimates on such solutions.


Quasilinear elliptic equation \(\Phi \)-Laplacian operator Positive solutions Asymptotic behaviour 

Mathematics Subject Classification

34B18 35B40 31C15 



  1. 1.
    Adams, R.A., Fournier, J.F.: Sobolev Spaces, 2nd edn. Academic Press, New York (2003)zbMATHGoogle Scholar
  2. 2.
    Bachar, I., Mâagli, H.: Existence and global asymptotic behavior of positive solutions for nonlinear problems on the half-line. J. Math. Anal. Appl. 416, 181–194 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bachar, I., Ben Othman, S., Mâagli, H.: Radial solutions for the \(p\)-Laplacian equation. Nonlinear Anal. Theory Methods Appl. 70(6), 2198–2205 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Benci, V., Fortunato, D., Pisani, L.: Soliton like solutions of a Lorentz invariant equation in dimension 3. Rev. Math. Phys. 10, 315–344 (1998)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ben Othman, S., Chemmam, R., Mâagli, H.: Asymptotic behavior of ground state solutions for \(p\)-Laplacian problems. J. Math. 2013, 1–7 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brezis, H., Kamin, S.: Sublinear elliptic equations in \( \mathbb{R} ^{n}\). Manuscr. Math. 74, 87–106 (1992)CrossRefGoogle Scholar
  7. 7.
    Brezis, H., Oswald, L.: Remarks on sublinear elliptic equations. Nonlinear Anal. 10, 55–64 (1986)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Calzolari, E., Filippucci, R., Pucci, P.: Existence of radial solutions for the \(p\)-Laplacian elliptic equations with weights. Am. Inst. Math. Sci. J. 15(2), 447–479 (2006)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Carvalho, M.L.M., Goncalves, Jose V.A., da Silva, E.D.: On quasilinear elliptic problems without the Ambrosetti–Rabinowitz condition. J. Math. Anal. Appl. 426, 466–483 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chemmam, R., Dhifli, A., Mâagli, H.: Asymptotic behavior of ground state solutions for sublinear and singular nonlinear Dirichlet problem. Electron. J. Differ. Equ. 88, 1–12 (2011)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Dacorogna, B.: Introduction to the Calculus of Variations. ICP, London (2004)CrossRefGoogle Scholar
  12. 12.
    Fuchs, M., Li, G.: Variational inequalities for energy functionals with nonstandard growth conditions. Abstr. Appl. Anal. 3, 41–64 (1998)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fuchs, M., Osmolovski, V.: Variational integrals on Orlicz–Sobolev spaces. Z. Anal. Anwend. 17, 393–415 (1998)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fukagai, N., Ito, M., Narukawa, K.: Positive solutions of quasilinear elliptic equations with critical Orlicz–Sobolev nonlinearity on \( \mathbb{R} ^{N}\). Funkc. Ekvac. 49(2), 235–267 (2006)CrossRefGoogle Scholar
  15. 15.
    Fukagai, N., Narukawa, K.: Nonlinear eigenvalue problem for a model equation of an elastic surface. Hiroshima Math. J. 25, 19–41 (1995)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Fukagai, N., Narukawa, K.: On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems. Ann. Mat. Pura Appl. 186, 539–564 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Karamata, J.: Sur un mode de croissance régulière. Théorèmes fondamentaux. Bull. Soc. Math. Fr. 61, 55–62 (1933)CrossRefGoogle Scholar
  18. 18.
    Karls, M., Mohammed, A.: Integrability of blow-up solutions to some non-linear differential equations. Electron. J. Differ. Equ. 2004, 1–8 (2004)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Khamessi, B.: Asymptotic behavior of ground state solutions of a non-linear Dirichlet problem. Complex Var. Elliptic Equ. (2017). CrossRefzbMATHGoogle Scholar
  20. 20.
    Lazer, A.C., Mckenna, P.J.: On a singular nonlinear elliptic boundary value problem. Proc. Am. Math. Soc. 111, 721–730 (1991)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Mâagli, H.: Asymptotic behavior of positive solution of a semilinear Dirichlet problem. Nonlinear Anal. 74, 2941–2947 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Maric, V.: Regular Variation and Differential Equations. Lecture Notes in Mathematics, vol. 1726. Springer, Berlin (2000)CrossRefGoogle Scholar
  23. 23.
    Masmoudi, S., Zermani, S.: Existence and asymptotic behavior of solutions to nonlinear radial \(p-\)Laplacian equations. Electron. J. Differ Equ. 2015, 1–12 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Pucci, P., García-Huidobro, M., Manásevich, R., Serrin, J.: Qualitative properties of ground states for singular elliptic equations with weights. Annali di Matematica Pura ed Applicata IV 185, S205–S243 (2006)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Radulescu, V.: Nonlinear elliptic equations with variable exponent old and new. Nonlinear Anal. 121, 336–369 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Rao, M.N., Ren, Z.D.: Theory of Orlicz Spaces. Marcel Dekker, New York (1985)Google Scholar
  27. 27.
    Růžička, M.: Electrorheological Fluids, Modeling and Mathematical Theory. Lecture Notes in Mathematics. Springer, Berlin (2000)CrossRefGoogle Scholar
  28. 28.
    Reichel, W., Walter, W.: Radial solutions of equations and inequalities involving the \(p\)-Laplacian. J. Inequal. Appl. 1, 47–71 (1997)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Resnick, S.I.: Extreme Values, Regular Variation, and Point Processes. Springer, New York (1987)CrossRefGoogle Scholar
  30. 30.
    Santos, C.A., Zhou, J., Santos, J.A.: Necessary and sufficient conditions for existence of blow-up solutions for elliptic problems in Orlicz–Sobolev spaces. Math. Nachr. 291, 160–177 (2018)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Seneta, R.: Regular Varying Functions. Lectures Notes in Mathematics. Springer, Berlin (1976)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Abdelwaheb Dhifli
    • 1
    Email author
  • Rym Chemmam
    • 2
  • Syrine Masmoudi
    • 2
  1. 1.Institut préparatoire aux etudes d’ingénieurs d’el Manar, LR10ES09 Modélisation mathématique, Analyse harmonique et théorie du potentielUniversité Tunis El ManarTunisTunisie
  2. 2.Faculté des sciences de Tunis, LR10ES09 Modélisation mathématique, Analyse harmonique et théorie du potentielUniversité Tunis El ManarTunisTunisie

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