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Mathematical Notes

, Volume 105, Issue 3–4, pp 404–424 | Cite as

Multiplicity Results for the Biharmonic Equation with Singular Nonlinearity of Super Exponential Growth in ℝ4

  • K. SaoudiEmail author
  • M. KratouEmail author
  • E. Al ZahraniEmail author
Article
  • 21 Downloads

Abstract

We consider the following elliptic problem of exponential-type growth posed in an open bounded domain with smooth boundary B1 (0) ⊂ ℝ4: \((P_\lambda)\begin{cases}\Delta^{2}u = \lambda(u^{-\delta}+h(u)e^{u^{\alpha}}),\;\;u>0\;in\;B_{1}(0),\\\;\;\;\;\;u=\Delta{u}=0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;on\;\partial{B}_{1}(0).\end{cases}\) Here Δ2(.):= −Δ(−Δ)(.) denotes the biharmonic operator, 1 < α < 2, 0 < δ < 1, λ > 0, and h(t) is assumed to be a smooth “perturbation” of \({e^{{t^\alpha }}}\) as t→∞ (see (H1)–(H4) below). We employ variational methods in order to show the existence of at least two distinct (positive) solutions to the problem (Pλ) in \({H^2} \cap H_0^1({B_1}(0))\).

Keywords

biharmonic equation multiple solutions super exponential growth Dirichlet boundary conditions 

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References

  1. 1.
    D. R. Adams, “A sharp inequality of J. Moser for higher order derivatives,” Ann. Math. 128, 385–398 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    A. Ambrosetti, H. Brezis and G. Cerami, “Combined effects of concave and convex nonlinearities in some elliptic problems,” J. Funct. Anal. 122, 519–543 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    H. Ansari, S. M. Vaezpour, and M. Hesaaraki, “Existence of positive solution for nonlocal singular fourth order Kirchhoff equation with Hardy potential,” Positivity 21 (4), 1545–1562 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    F. Bernis, J. Garcia Azorero, and I. Peral, “Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order,” Adv. Differential Equations 1 (2), 219–240 (1996).MathSciNetzbMATHGoogle Scholar
  5. 5.
    M. Badiale and Tarantello, “Existence and multiplicity for elliptic problems with critical growth and discontinuous nonlinearities,” Nonlinear Analysis The., Metho. and Appl., 29, 639–677 (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    M. M. Coclite and G. Palmieri, “On a singular nonlinear Dirichlet problem,” Comm. Partial Differential Equations 14 (10), 1315–1327 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    M. G. Crandall, P. H. Rabinowitz, and L. Tartar, “On a Dirichlet problem with a singular nonlinearity,” Comm. Partial Differential Equations 2 (2), 193–222 (1977).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    J. I. Díaz, J. Hernández, and J. M. Rakotoson, “On very weak positive solutions to some semilinear elliptic problems with simultaneous singular nonlinear and spatial dependence terms,” Milan J. Math. 79 (1), 233–245 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    R. Dhanya, J. Giacomoni, S. Prashanth, and K. Saoudi, “Multiplicity results for elliptic equations with singular nonlinearity of super exponential growth in ℝ2,” Advances in Differential Equations, 17 (3–4), (March/April) (2012).Google Scholar
  10. 10.
    M. Ghergu, “A biharmonic equation with singular nonlinearity,” Proceedings of the Edinburgh Mathematical Society, 55, 155–166 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    A. Ghanmi and K. Saoudi, “The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator,” Fractional Differential Calculus, 6 (2), 201–217 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    A. Ghanmi and K. Saoudi, “A multiplicity results for a singular problem involving the fractional p-Laplacian operator,” Complex Var. Elliptic Equ. 61, 1199–1216 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    J. Giacomoni and K. Saoudi, “Multiplicity of positive solutions for a singular and critical problem,” Nonlinear Anal. 71 (9), 4060–4077 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    N. Ghoussoub and D. Preiss, “A general mountain pass principle for locating and classifying critical points,” Ann. Inst. H. Poincaré Anal. NonLinéaire 6 (5) 321–330 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    H. Grunau and G. Sweers, “Classical solutions for some higher order semilinear elliptic equations under weak growth conditions,” Nonlin. Anal. 28, 799–807 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Y. Haitao, “Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem,” J. Differential Equations, 189 (2), 487–512 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    L. G. Hernandez and Y. Choi, “Existence of solutions in a singular biharmonic nonlinear problem,” Proc. Edinburgh Math. Soc. 2 (3), 537–546 (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    N. Hirano, C. Saccon, and N. Shioji, “Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities,” Adv. Differential Equations 9 (1–2), 197–220 (2004).MathSciNetzbMATHGoogle Scholar
  19. 19.
    B. S. Kaur and K. Sreenadh, “Multiple positive solutions for a singular biharmonic equation in ℝ4,” Complex Var. Elliptic Equ. 54 (1), 9–21 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    J. Moser, “A sharp form of an inequality by N. Trudinger,” Indiana Univ. Math. J. 20, 1077–1092 (1970/1971).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    P. L. Lions, “The concentration compactness principale in calculus of variations Part 1,” Revista Mathematica Iberoamericana 1, 185–201 (1985).Google Scholar
  22. 22.
    W. M. Ni and I. Takagi, “On the shape of Least-Energy solutions to a semilinear Neumann problem,” Comm. Pure and Applied Math. XLIV (8/9), 819–851 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    W. V. Petryshyn and J. R. L. Webb, “Existence and multiplicity results for nonstandard semilinear biharmonic boundary value problems,” Nonlinear Anal. 28 (6), 965–981 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    K. Saoudi, “Existence and non-existence for a singular problem with variables potentials,” Electronic Journal of Differential Equations 2017 (291), 1–9 (2017).MathSciNetzbMATHGoogle Scholar
  25. 25.
    K. Saoudi and M. Kratou, “Existence of multiple solutions for a singular and quasilinear equation,” Complex Var. Elliptic Equ. 60 (7), 893–925 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    G. Stampacchia, Equations elliptiques du second ordre á coefficients discontinues (Les Presses de l’Université de Montréal, 1966).zbMATHGoogle Scholar
  27. 27.
    P. Takáč, “On the Fredholm Alternative for the p-Laplacian at the first eigenvalue,” Indiana Univ. J. 51 (1), 187–237 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    X. Z. Zeng and Y. B. Deng, “Existence of multiple solutions for a semilinear biharmonic equation with critical exponent,” (Chinese) Acta Math. Sci. Ser. A Chin. Ed. 20 (4), 547–554 (2000).MathSciNetzbMATHGoogle Scholar
  29. 29.
    X. Z. Zeng and L. M. L. Huang, “Existence of multiple solutions for critical semilinear polyharmonic equations in ℝN,” Appl. Math. J. Chinese Univ. Ser. A 16 (4), 414–420 (2001) [in Chinese].MathSciNetzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Department of MathematicsImam Abdulrahman Bin Faisal UniversityDammamKingdom of Saudi Arabia
  2. 2.Basic and Applied Scientific Research CenterImam Abdulrahman Bin Faisal UniversityDammamKingdom of Saudi Arabia

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