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Existence of Positive Solutions to Weighted Linear Elliptic Equations Under Double Exponential Nonlinearity Growth

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Abstract

In this article, a weighted problem under boundary Dirichlet condition in the unit ball of \({\mathbb {R}}^{2}\) is considered. The non-linearity of the equation is assumed to have double exponential growth in view of Trudinger–Moser type inequalities. It is proved that there is a nontrivial positive weak solution to this equation. In the critical case, the compactness condition is not satisfied but a suitable asymptotic condition is used to avoid the non-compactness levels to the energy functional.

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References

  1. Adimurthi, A.: Existence results for the semilinear Dirichlet problem with critical growth for the n-Laplacian. Houst. J. Math. 7, 285–298 (1991). (MR 1079983|Zbl 0732.35028)

    MATH  Google Scholar 

  2. Adimurthi, A., Sandeep, K.: A singular Moser–Trudinger embedding and its applications. Nonlinear Differ. Equ. Appl. 13(5–6), 585–603 (2007). https://doi.org/10.1007/s00030-006-4025-9

    Article  MathSciNet  MATH  Google Scholar 

  3. Alama, S., Li, Y.Y.: Existence of solutions for semilinear elliptic equations with indefinite linear part. J. Differ. Equ. 96, 89–115 (1992)

    Article  MathSciNet  Google Scholar 

  4. Alves, C., de Freitas, L.: Multiplicity of nonradial solutions for a class of quasilinear equations on annulus with exponential critical growth. Topol. Methods Nonlinear Anal. 39(2), 243–262 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Alves, C.O., do Ó, J.M., Miyagaki, O.H.: On nonlinear perturbations of a periodic elliptic problem in \({\mathbb{R}}^{2}\) involving critical growth. Nonlinear Anal. 56, 781–791 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Aris, R.: The Mathematical Theory of Diffusion and Reaction in Permeable Catalyst, vol. 2. Clarendon Press, Oxford (1975)

    MATH  Google Scholar 

  8. Astrita, G., Marrucci, G.: Principles of Non-Newtonian Fluid Mechanics. McGraw-Hill, New York (1974)

    Google Scholar 

  9. Bornemann, F., März, T.: Fast image inpainting based on coherence transport. J. Math. Imaging Vis. 28, 259–278 (2007)

    Article  MathSciNet  Google Scholar 

  10. Caglioti, E., Lions, P.L., Marchioro, C., Pulvirenti, M.: A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Commun. Math. Phys. 143(3), 501–525 (1992). https://doi.org/10.1007/BF02099262

    Article  MathSciNet  MATH  Google Scholar 

  11. Caglioti, E., Lions, P.L., Marchioro, C., Pulvirenti, M.: A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description, II. Commun. Math. Phys. 174(2), 229–260 (1995). https://doi.org/10.1007/BF02099602

    Article  MathSciNet  MATH  Google Scholar 

  12. Calanchi, M., Ruf, B.: On a Trudinger–Moser type inequalities with logarithmic weights. J. Differ. Equ. 3, 258–263 (2015). https://doi.org/10.1016/j.jde.2014.11.019

    Article  MathSciNet  MATH  Google Scholar 

  13. Calanchi, M., Ruf, B.: Trudinger–Moser type inequalities with logarithmic weights in dimension N. Nonlinear Anal. Ser. A Theory Methods Appl. 121, 403–411 (2015). https://doi.org/10.1016/j.na.2015.02.001

    Article  MathSciNet  MATH  Google Scholar 

  14. Calanchi, M., Ruf, B.: Weighted Trudinger–Moser inequalities and applications. Bull. South Ural State Univ. Ser. Math. Model. Program. Comput. Softw. 8(3), 42–55 (2015). https://doi.org/10.14529/mmp150303

    Article  MATH  Google Scholar 

  15. Calanchi, M., Ruf, B., Sani, F.: Elliptic equations in dimension 2 with double exponential nonlinearities. Nonlinear Differ. Equ. Appl. NoDea 24, 29 (2017). https://doi.org/10.1007/s00030-017-0453-y

    Article  MathSciNet  MATH  Google Scholar 

  16. Calanchi, M., Terraneo, E.: Non-radial Maximizers For Functionals With Exponential Non- linearity in \({\mathbb{R}}^{2}\). Adv. Nonlinear Stud. 5, 337–350 (2005). https://doi.org/10.1515/ans-2005-0302

    Article  MathSciNet  MATH  Google Scholar 

  17. Chanillo, S., Kiessling, M.: Rotational symmetry of solutions of some nonlinear problems in statistical mechanics and in geometry. Commun. Math. Phys. 160(2), 217–238 (1994). https://doi.org/10.1007/BF02103274

    Article  MathSciNet  MATH  Google Scholar 

  18. de Figueiredo, D.G., Miyagaki, O.H., Ruf, B.: Elliptic equations in \({\mathbb{R}}^{2}\) with nonlinearities in the critical growth range. Calc. Var. Partial Differ. Equ. 3(2), 139–153 (1995). https://doi.org/10.1007/BF01205003

    Article  MATH  Google Scholar 

  19. de Figueiredo, D.G., do Ó, J.M., Ruf, B.: On an inequality. In: Trudinger, N., Moser, J. (Eds.) Related Elliptic Equations, Comm. Pure Appl. Math. LV, pp. 135–152 (2002)

  20. Ding, W.Y., Ni, W.M.: On the existence of positive entire solutions of a semilinear elliptic equation. Arch. Ration. Math. Anal. 31, 283–308 (1986)

    Article  MathSciNet  Google Scholar 

  21. do Ó, J.M., Ruf, B.: On a Schrödinger equation with periodic potential and critical growth in \({\mathbb{R}}^{2}\). NoDEA Nonlinear Differ. Equ. Appl. 13, 167–192 (2006)

    Article  Google Scholar 

  22. do Ó, J.M., Ruf, B.: On a Schrödinger equation with periodic potential and critical growth in \({\mathbb{R}}^{2}\). Commun. Pure Appl. Math. 2, 135–152 (2002)

    Google Scholar 

  23. Drabek, P., Kufner, A., Nicolosi, F.: Quasilinear Elliptic Equations with Degenerations and Singularities. Walter de Gruyter, Berlin (1997). https://doi.org/10.1515/9783110804775

    Book  MATH  Google Scholar 

  24. Garain, P.: On a degenerate singular elliptic problem. arXiv:1803.02102

  25. Garain, P., Mukherjee, T.: On a class of weighted p-Laplace equation with singular nonlinearity. Mediterr. J. Math. 17, 110 (2020). https://doi.org/10.1007/s00009-020-01548-w

    Article  MathSciNet  MATH  Google Scholar 

  26. Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear potential theory of degenerate elliptic equations, Oxford Mathematical Monographs, p. vi+363. The Clarendon Press, Oxford University Press, Oxford Science Publications, New York (1993)

    MATH  Google Scholar 

  27. Jeanjean, L.: Solutions in spectral gaps for a nonlinear equation of Schrödinger type. J. Differ. Equ. 112, 53–80 (1994)

    Article  Google Scholar 

  28. Kiessling, M.K.-H.: Statistical mechanics of classical particles with logarithmic interactions. Commun. Pure Appl. Math. 46, 27–56 (1993). https://doi.org/10.1002/cpa.3160460103

    Article  MathSciNet  MATH  Google Scholar 

  29. Kryszewski, W., Szulkin, A.: Generalized linking theorem with an application to semilinear Schrödinger equation. Adv. Differ. Equ. 3, 441–472 (1998)

    MATH  Google Scholar 

  30. Kufner, A.: Weighted Sobolev Spaces. Wiley, New York (1985). https://doi.org/10.1112/blms/18.2.220

    Book  MATH  Google Scholar 

  31. Lions, P.L.: The concentration-compactness principle in the calculus of variations, part 1. Rev. Iberoam. 11, 185–201 (1985)

    Google Scholar 

  32. Liouville, J.: Sur l’ equation aux derivées partielles. J. Math. Pures Appl. 18, 71–72 (1853)

    Google Scholar 

  33. Masmoudi, N., Sani, F.: Trudinger–Moser inequalities with the exact growth condition in \(\mathbb{R^{N}}\) and applications. Commun. Partial Differ. Equ. 40(8), 1408–1440 (2015). https://doi.org/10.1080/03605302.2015.1026775

    Article  MATH  Google Scholar 

  34. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990). https://doi.org/10.1109/34.56205

    Article  Google Scholar 

  35. Rabinowitz, P.H.: Mini–max methods in critical point theory with applications to differential equations. In: CBMS Regional Conference Series in Math, no. 65. AMS, Providence (1986)

  36. Tarantello, G.: Condensate solutions for the Chern–Simons–Higgs theory. J. Math. Phys. 37, 3769–3796 (1996). https://doi.org/10.1063/1.531601

    Article  MathSciNet  MATH  Google Scholar 

  37. Tarantello, G.: Analytical aspects of Liouville-type equations with singular sources. In: Chipot, M., Quittner, P. (eds.) Handbook of Differential Equations, pp. 491–592. Elsevier, North Holland (2004)

    MATH  Google Scholar 

  38. Volquer, R.E.: Nonlinear flow in porus media by finite elements. ASCE Proc. J. Hydraul. Div. Proc Am. Soc. Civ. Eng. 95, 2093–2114 (1969)

    Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, Faculty of Arts and Science in Qilwah, Al Baha University, Baha, Kingdom of Saudi Arabia

    Saber Kharrati

  2. Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Tunis, Tunisia

    Saber Kharrati & Rached Jaidane

Authors
  1. Saber Kharrati
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  2. Rached Jaidane
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Correspondence to Saber Kharrati.

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Communicated by Amin Esfahani.

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Kharrati, S., Jaidane, R. Existence of Positive Solutions to Weighted Linear Elliptic Equations Under Double Exponential Nonlinearity Growth. Bull. Iran. Math. Soc. 48, 993–1021 (2022). https://doi.org/10.1007/s41980-021-00559-x

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