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Existence and Multiplicity Results for an Elliptic Problem Involving Cylindrical Weights and a Homogeneous Term \(\mu \)

  • R. B. Assunção
  • O. H. Miyagaki
  • L. C. Paes-Leme
  • B. M. RodriguesEmail author
Article
  • 21 Downloads

Abstract

We consider the following elliptic problem
$$\begin{aligned} \left\{ \begin{array}{lll} -{\text {div}}\left( \dfrac{\left| \nabla u\right| ^{p-2} \nabla u}{\left| y\right| ^{ap}}\right) = \mu \dfrac{\left| u\right| ^{p-2} u}{\left| y\right| ^{p(a+1)}}+ h(x) \dfrac{\left| u\right| ^{q-2} u}{\left| y\right| ^{bq}} + f(x,u) &{}&{} \text{ in } \ \Omega , \\ u = 0 &{}&{} \text{ on } \ \partial \Omega ,\\ \end{array} \right. \end{aligned}$$
in an unbounded cylindrical domain
$$\begin{aligned} \Omega :=\{ (y,z)\in {\mathbb {R}}^{m+1}\times {\mathbb {R}}^{N-m-1} \ ; \ 0<A<\left| y\right|<B <\infty \}, \end{aligned}$$
where \(A,B\in {\mathbb {R}}_+\), \(p>1\), \(1\le m<N-p\), \(q:=\dfrac{Np}{N-p (a+1-b)},\) \(0\le \mu < {\overline{\mu }}:=\left( \dfrac{m+1-p(a+1)}{p}\right) ^p \), \(h\in L^{\frac{N}{q}}(\Omega )\cap L^{\infty }(\Omega )\) is a positive function and \(f: \Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a Carathéodory function with growth at infinity. Using the Krasnoselski’s genus and applying \({\mathbb {Z}}_2\) version of the Mountain Pass Theorem, we prove, under certain assumptions about f, that the above problem has infinite invariant solutions.

Keywords

Supercritical degenerate operator variational methods 

Mathematics Subject Classification

35B07 35J62 35J70 

Notes

Acknowledgements

O.H. Miyagaki received research grants from CNPq/Brazil Proc.304015/2014-8, FAPEMIG/Brazil CEX APQ-00063/15 and INCTMAT/CNPq/Brazil.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no competing interests.

References

  1. 1.
    Adams, A.R.: Sobolev Spaces. Academic Press, New York (1975)zbMATHGoogle Scholar
  2. 2.
    Ambrosetti, A., Brézis, H., Cerami, G.: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 2, 519–543 (1994)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Assunção, RB., dos Santos, WW., Miyagaki, OH.: Existence and multiplicity results on a class of quasilinear elliptic problems with cylindrical singularities involving multiple critical exponents. arXiv: 1506.09162 [math.AP] (2015)
  4. 4.
    Assunção, RB., dos Santos, WW., Miyagaki, OH.: Quasilinear elliptic problems with cylindrical singularities and multiple critical nonlinearities: existence, regularity, nonexistence. arXiv:1506.09152 [math.AP] (2015)
  5. 5.
    Assunção, R.B., Miyagaki, O.H., Rodrigues, B.M.: Existence and multiplicity of solutions for a supercritical elliptic problem in unbounded cylinders. Bound. Value Probl. 2017, 52 (2017)Google Scholar
  6. 6.
    Badiale, M., Serra, E.: Semilinear Elliptic Equations for Beginners. Existence Results via the Variational Approach. Springer, London (2011)CrossRefGoogle Scholar
  7. 7.
    Badiale, M., Tarantello, G.: A Sobolev-Hardy inequality whith applications to a nonlinear elliptic equation arising in astrophysics. Arch. Ration. Mech. Anal. 163, 259–293 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bartle, R.: The Elements of Integration and Lebesgue Measure. Wiley-Interscience, Hoboken (1995)CrossRefGoogle Scholar
  9. 9.
    Bhakta, M.: On the existence and breaking symmetry of the ground state solution of Hardy Sobolev type equations with weighted p-Laplacian. Adv. Nonlinear Stud. 3, 555–568 (2012)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bouchekif, M., El Mokhtar, M.E.M.O.: Nonhomogeneous elliptic equations with decaying cylindrical potential and critical exponent. Electron. J. Differ. Equ. 54, 1–10 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bouchekif, M., Matallah, A.: Multiple positive solutions for elliptic equations involving a concave term and critical Sobolev-Hardy exponent. Appl. Math. Lett. 22, 268–275 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cao, D., Han, P.: Solutions for semilinear elliptic equations with critical exponents and Hardy potential. J. Differ. Equ. 2, 521–537 (2004)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chen, J.: Multiple positive solutions for a class of nonlinear elliptic equations. J. Math. Anal. Appl. 2, 341–354 (2004)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Clapp, M., Szulkin, A.: A supercritical elliptic problem in a cylindrical shell. Prog. Nonlinear Differ. Equ. Appl. 85, 233–242 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology I: Physical Origins and Classical Methods. Springer, Berlin (1985)zbMATHGoogle Scholar
  16. 16.
    Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Fan, X.L., Zhao, Y.Z.: Linking and multiplicity results for the p-Laplacian on unbounded cylinders. J. Math. Anal. Appl. 260, 479–489 (2001)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ferrero, A., Gazzola, F.: Existence of solutions for singular critical growth semilinear elliptic equations. J. Differ. Equ. 2, 494–522 (2001)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Filippucci, R., Pucci, P., Robert, F.: On a p-Laplace equation with multiple critical nonlinearities. J. Math. Pures Appl. 9, 156–177 (2009)MathSciNetCrossRefGoogle Scholar
  20. 20.
    García, A., Peral, A.: Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term. Trans. Am. Math. Soc. 2, 877–895 (1991)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Gazzini, M., Musina, R.: On a Sobolev-type inequality related to the weighted p-Laplace operator. J. Math. Anal. Appl. 1, 99–111 (2009)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ghergu, M., Radulescu, V.: Singular elliptic problems with lack of compactness. Ann. Mat. Pura Appl. 4, 63–79 (2006)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ghergu, M., Radulescu, V.: Nonlinear PDEs, Mathematical models in biology, chemistry and population genetics. Springer, Berlin (2012)zbMATHGoogle Scholar
  24. 24.
    Ghoussoub, N., Robert, F.: Concentration estimates for Emden-Fowler equations with boundary singularities and critical growth. Int. Math. Res. Pap. 21867, 1–85 (2006)Google Scholar
  25. 25.
    Ghoussoub, N., Yuan, C.: Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. Trans. Am. Math. Soc. 12, 5703–5743 (2000)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Hsu, T.S.: Multiplicity results for p -Laplacian with critical nonlinearity of concave-convex type and sign-changing weight functions. Abstr. Appl. Anal. 2009, 1–24 (2009)CrossRefGoogle Scholar
  27. 27.
    Hsu, T.S.: Multiple positive solutions for a quasilinear elliptic problem involving critical Sobolev-Hardy exponents and concave-convex nonlinearities. Nonlinear Anal. 12, 3934–3944 (2011)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Hsu, T.S., Lin, H.L.: Multiple positive solutions for singular elliptic equations with concave-convex nonlinearities and sign-changing weights. Bound. Value Probl. 2009, 1–17 (2009)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Hsu, T.S., Lin, H.L.: Multiple positive solutions for singular elliptic equations with weighted Hardy terms and critical Sobolev–Hardy exponents. Proc. Roy. Soc. Edinb. Sect. A. 3, 617–633 (2010)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Hsu, T.S., Lin, H.L.: Multiplicity of positive solutions for weighted quasilinear elliptic equations involving critical Hardy-Sobolev exponents and concave-convex nonlinearities. Abstr. Appl. Anal. 1–1 (2012)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Kang, D., Peng, S.: Solutions for semilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy potential. Appl. Math. Lett. 10, 1094–1100 (2005)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Kavian, O.: Introduction à la Théorie des Points Critiques. Springer, Paris (1993)zbMATHGoogle Scholar
  33. 33.
    Kesavan, S.: Nonlinear Functional Analysis. A First Course. Hindustan Book Agency, Gurgaon (2004)CrossRefGoogle Scholar
  34. 34.
    Lao, Y.S.: Nonlinear p-Laplacian problems on unbounded domains. Proc. Am. Math. Soc. 115, 1037–1045 (1992)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Lions, L.P.: Symmétrie et compacité dans les espaces sobolev. J. Funct. Anal. 49, 315–334 (1982)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Palais, S.R.: The principle of symmetric criticality. Commun. Math. Phys. 69, 19–30 (1979)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Rabinowtitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. Amer. Math. Soc, Providence, Rhode Island (1986)Google Scholar
  38. 38.
    Secchi, S., Smets, D., Willem, M.: Remarks on a Hardy-Sobolev inequality. C. R. Math. Acad. Sci. 10, 811–815 (2003)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Sun, X.: p-Laplace equations with multiple critical exponents and singular cylindrical potential. Acta Math. Sci. Ser. B Engl. Ed. 4, 1099–1112 (2013)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Tarantello, G.: On nonhomogeneous elliptic equations involving critical Sobolev exponent. Ann. Inst. H. Poincaré Anal. Non Linéaire 3, 281–304 (1992)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Wang, L., Wei, Q., Kang, D.: Multiple positive solutions for p-Laplace elliptic equations involving concave-convex nonlinearities and a Hardy-type term. Nonlinear Anal. 2, 626–638 (2011)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Willem, M.: Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications. Birkh\(\ddot{\text{a}}\)user, Boston (1996)Google Scholar
  43. 43.
    Xuan, B., Wang, J.: Existence of a nontrivial weak solution to quasilinear elliptic equations with singular weights and multiple critical exponents. Nonlinear Anal. 3649–3658 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • R. B. Assunção
    • 1
  • O. H. Miyagaki
    • 2
  • L. C. Paes-Leme
    • 3
  • B. M. Rodrigues
    • 3
    Email author
  1. 1.Departamento de MatemáticaUniversidade Federal de Minas GeraisBelo HorizonteBrazil
  2. 2.Departamento de MatemáticaUniversidade Federal de Juiz de ForaJuiz de ForaBrazil
  3. 3.Departamento de MatemáticaUniversidade Federal de Ouro PretoOuro PretoBrazil

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