Skip to main content
SpringerLink
Log in

A type of Brézis–Oswald problem to \(\Phi \)-Laplacian operator with strongly-singular and gradient terms

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

It is established existence of minimal \(W_{{\textit{loc}}}^{1,\Phi }(\Omega )\)-solutions on some appropriated set for the quasilinear elliptic problem

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta _\Phi u= \lambda f(x,u)+\mu h(x,u, \nabla u )\,\text{ in }\,\Omega ,\\ u>0\,\text{ in }\,\Omega ,\quad u=0\,\text{ on }\,\partial \Omega , \end{array}\right. \end{aligned}$$

where f may have indefinite sign and behaves in a strongly singular way at \(u=0\), h has sublinear growth, \(\lambda >0\) and \(\mu \ge 0\) are real parameters. The main results improve the classical Brézis–Oswald’s result to Orlicz–Sobolev setting in the presence of strongly-singular nonlinearities as well as gradient terms. Our approach is based on a new comparison principle for \(W_{{\textit{loc}}}^{1,\Phi }(\Omega )\)-sub and super solutions to the \(\Phi \)-Laplacian operator established here, truncation arguments, \(L^{\infty }(\Omega )\) estimates and a generalized Galerkin method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abdellaoui, B., Dall’Aglio, A., Peral, I.: Some remarks on elliptic problems with critical growth in the gradient. J. Differ. Equ. 222(1), 21–62 (2006)

  2. Adams, R.A., Fournier, J.F.: Sobolev Spaces. Academic Press, New York (2003)

    MATH  Google Scholar 

  3. Brézis, H., Oswald, L.: Remarks on sublinear elliptic equations. Nonlinear Anal. 10, 55–64 (1986)

    Article  MathSciNet  Google Scholar 

  4. Bueno, H., Ercole, G., Zumpano, A., Ferreira, W.M.: Positive solutions for the p-Laplacian with dependence on the gradient. Nonlinearity 25(4), 1211–1234 (2012)

    Article  MathSciNet  Google Scholar 

  5. Carvalho, M.L.M., Gonçalves, J.V.: Multivalued equations on a bounded domain via minimization on Orlicz–Sobolev spaces. J. Convex Anal. 21(1), 201–218 (2014)

    MathSciNet  MATH  Google Scholar 

  6. Carvalho, M.L., Goncalves, J.V., da Silva, E.D.: On quasilinear elliptic problems without the Ambrosetti Rabinowitz condition. J. Anal. Mat. Appl. 426, 466–483 (2015)

    Article  MathSciNet  Google Scholar 

  7. Carvalho, M.L., Goncalves, J.V., Silva, E.D., Santos, C.A.: A type of Brézis–Oswald problem to the \(\Phi \)-Laplacian operator with very singular term. Milan J. Math. 86, 53–80 (2018)

    Article  MathSciNet  Google Scholar 

  8. Carvalho, M.L., Goncalves, J.V., Santos, C.A.: About positive \(W_{loc}^{1,\Phi }(\Omega )\)-solutions to quasilinear elliptic problems with singular semilinear term. Topol. Methods Nonlinear Anal. 53(2), 491–517 (2019)

    MathSciNet  Google Scholar 

  9. Clément, P., de Pagter, B., Sweers, G., de Thélin, F.: Existence of solutions to a semilinear elliptic system through Orlicz–Sobolev spaces. Mediterr. J. Math. 1(3), 241–267 (2004)

    Article  MathSciNet  Google Scholar 

  10. Crandall, M., Rabinowitz, P.H., Tartar, L.: On a Dirichlet problem with a singular nonlinearity. Commun. Partial Differ. Equ. 2(2), 193–222 (1977)

    Article  MathSciNet  Google Scholar 

  11. Cuesta, M.: Existence results for quasilinear problems via ordered sub and supersolutions. Ann. Fac. Sci. Toulouse 6, 591–608 (1997)

    Article  MathSciNet  Google Scholar 

  12. de Figueiredo, D.G., Gossez, J.-P., Quoirin, H.R., Ubilla, P.: Elliptic equations involving the p-Laplacian and a gradient term having natural growth. Calc. Var. Partial Differ. Equ. 56(2), 32–56 (2017)

    Article  Google Scholar 

  13. Díaz, J.I., Saa, J.E.: Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires. C.R.A.S. de Paris t. 305, Serie I, pp. 521–524 (1987)

  14. Dong, G., Fang, X.: Differential equations of divergence form in separable Musielak–Orlicz–Sobolev spaces. Bound Value Probl. 106, 1–19 (2016)

    MathSciNet  MATH  Google Scholar 

  15. Faraci, F., Pugliese, D.: A singular semilinear problem with dependence on the gradient. J. Differ. Equ. 260, 3327–3349 (2016)

    Article  MathSciNet  Google Scholar 

  16. Faria, L.F.O., Miyagaki, O.H., Motreanu, D.: Comparison and positive solutions for problems with \((p, q)\)-Laplacian and a convection term. Proc. Edinb. Math. Soc. 57, 687–698 (2014)

    Article  MathSciNet  Google Scholar 

  17. Fukagai, N., Narukawa, K.: On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problem. Ann. Mat. Pura Appl. 186(3), 539–564 (2007)

    Article  MathSciNet  Google Scholar 

  18. Fukagai, N., Ito, M., Narukawa, K.: Positive solutions of quasilinear elliptic equations with critical Orlicz–Sobolev nonlinearity on R\(^{{\rm N}}\). Funkcialaj Ekvacioj 49, 235–267 (2006)

    Article  MathSciNet  Google Scholar 

  19. Gonçalves, J.V., Carvalho, M.L., Santos, C.A.: Quasilinear elliptic systems with convex-concave singular terms and \(\Phi \)-Laplacian operator. Differ. Integral Equ. 31(3/4), 231–256 (2018)

    MathSciNet  MATH  Google Scholar 

  20. Gossez, J.P.: Orlicz–Sobolev spaces and nonlinear elliptic boundary value problems. In: Nonlinear Analysis, Function Spaces and Applications (Proc. Spring School, Horni Bradlo, 1978), Teubner, Leipzig, pp. 59–94 (1979)

  21. Gossez, J.-P.: Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients. Trans. Am. Math. Soc. 190, 163–205 (1974)

    Article  MathSciNet  Google Scholar 

  22. Iturriaga, L., Lorca, S., Sánchez, J.: Existence and multiplicity results for the p-Laplacian with a p-gradient term. NoDEA Nonlinear Differ. Equ. Appl. 15, 729–743 (2008)

    Article  MathSciNet  Google Scholar 

  23. Iturriaga, L., Lorca, S., Ubilla, P.: A quasilinear problem without the Ambrosetti–Rabinowitz-type condition. Proc. Roy. Soc. Edinb. Sect. A 140(2), 391–398 (2010)

    Article  MathSciNet  Google Scholar 

  24. Jeanjean, L., Quoirin, H.R.: Multiple solutions for an indefinite elliptic problem with critical growth in the gradient. Proc. Am. Math. Soc. 144(2), 575–586 (2016)

    Article  MathSciNet  Google Scholar 

  25. Jeanjean, L., Sirakov, B.: Existence and multiplicity for elliptic problems with quadratic growth in the gradient. Commun. Partial Differ. Equ. 38, 244–264 (2013)

    Article  MathSciNet  Google Scholar 

  26. 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  Google Scholar 

  27. Lazer, A.C., McKenna, P.J.: On a singular nonlinear elliptic boundary-value problem. Proc. Am. Math. Soc. 111(1), 721–730 (1991)

    Article  MathSciNet  Google Scholar 

  28. Liu, Z., Motreanu, D., Zeng, S.: Positive solutions for nonlinear singular elliptic equations of p-Laplacian type with dependence on the gradient. Calc. Var. 58, 28 (2019)

    Article  MathSciNet  Google Scholar 

  29. Loc, N.H.: Schmitt, applications of sub-supersolution theorems to singular nonlinear elliptic problems. Adv. Nonlinear Stud. 11, 493–524 (2011)

    Article  MathSciNet  Google Scholar 

  30. Mokrane, A., Murat, F.: Lewy–Stampacchia’s inequality for a Leray–Lions operator with natural growth in the gradient. Boll. Unione Mat. Ital. 7, 55–85 (2014)

  31. Rao, M.N., Ren, Z.D.: Theory of Orlicz Spaces. Marcel Dekker, New York (1985)

    Google Scholar 

  32. Serrin, J.: The solvability of boundary value problems. In: Mathematical Developments Arising from Hilbert Problems, (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill.,1974), pp. 507–524. Proc. Sympos. Pure Math., Vol. XXVIII. American Mathematics Society, Providence, RI (1976)

  33. Sun, Y., Zhang, D.: The role of the power 3 for elliptic equations with negative exponents. Calc. Var. Partial Differ. Equ. 49, 909–922 (2014)

    Article  MathSciNet  Google Scholar 

  34. Tan, Z., Fang, F.: Orlicz–Sobolev versus Hölder local minimizer and multiplicity results for quasilinear elliptic equations. J. Math. Anal. Appl. 402, 348–370 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade Federal de Goiás, Goiânia, GO, 74001-970, Brazil

    M. L. Carvalho & E. D. Silva

  2. Departamento de Matemática, Universidade de Brasília, Brasília, DF, 70910-900, Brazil

    J. V. Goncalves & C. A. P. Santos

Authors
  1. M. L. Carvalho
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. V. Goncalves
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. D. Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. C. A. P. Santos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. A. P. Santos.

Additional information

Communicated by Manuel del Pino.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The E. D. Silva was also partially supported by CNPq with grants 429955/2018-9 and 309026/2020-2

The C. A. Santos was supported by CNPq/Brazil with grant 311562/2020 - 5.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carvalho, M.L., Goncalves, J.V., Silva, E.D. et al. A type of Brézis–Oswald problem to \(\Phi \)-Laplacian operator with strongly-singular and gradient terms. Calc. Var. 60, 195 (2021). https://doi.org/10.1007/s00526-021-02075-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-021-02075-6

Mathematics Subject Classification

Search

Navigation