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Weighted \(p(\cdot )\)-Laplacian problem with nonlinear singular terms

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Abstract

In the present work, we prove an existence and uniqueness result of solutions to a quasilinear elliptic problem with nonlinear singular terms in the weighted Sobolev space. The equation that we consider is the following

$$\begin{aligned} -\Delta _{p(\cdot )}^{\omega } u+\beta (u)=\frac{f(x)}{u^{\alpha }}, \end{aligned}$$

where \(\alpha \ge 1\), \(\beta \) is a continuous non decreasing surjective real function on \({\mathbb {R}}\), f is a nonnegative function belonging to the Lebesgue space \(L^{m}(\Omega )\) and \(m\ge 1\).

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References

  1. Arcoya, D., Ruiz, D.: The Ambrosetti–Prodi problem for the p-Laplace operator. Commun. Partial Differ. Equ. 31(6), 849–865 (2006)

    Article  MATH  Google Scholar 

  2. Azroul, E., Barbara, A., Benboubker, M.B., El Haiti, K.: Existence of entropy solutions for degenerate elliptic unilateral problems with variable exponents. Boletim da Sociedade Paranaense de Matemática 36(1), 79–99 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brandolini, B., Ferone, V., Messano, B.: Existence and comparison results for a singular semilinear elliptic equation with a lower order term. Ricerche mat. 63(Suppl 1), S3–S18 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bouhlal, A., Igbida, J., Talibi, H., El Hachimi, A.: Existence of solutions for unbounded elliptic equations with critical natural growth. Int. J. Differ. Equ. 4, 53–64 (2018)

    MathSciNet  Google Scholar 

  5. Chu, Y., Gao, W.: Existence of solutions to a class of quasilinear elliptic problems with nonlinear singular terms. Bound. Value Probl. 229(1), 1–8 (2013)

    MathSciNet  MATH  Google Scholar 

  6. El-Hadi, M.S.: On an anisotropic problem with singular nonlinearity having variable exponent. Ricerche mat. 66, 415–424 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fan, X.L., Zhao, D.: On the spaces of \(L^{p(x)}(\Omega )\) and \(W^{m,p(x)}(\Omega )\). J. Math. Anal. Appl. 263, 424–446 (2001)

  8. Hasto, P.: The \(p(x)\)-Laplacian and applications. J. Anal. 15, 53–62 (2007)

    MathSciNet  MATH  Google Scholar 

  9. Igbida, J., Bouhlal, A., Talibi, H., El Hachimi, A.: Borderline cases of degenerate elliptic equations having gradient and lower term. Int. J. Math. Comput. 28(4), 104–117 (2017)

    Google Scholar 

  10. Igbida, J., Bouhlal, A., Talibi, H., El Hachimi, A.: Unbounded elliptic equation with singular critical growth to the gradient. J. Nonlinear Syst. Appl. 151–158 (2017)

  11. Kaushik, B., Prashanta, G.: Nonexistence results for weighted \(p\)-Laplace equations with singular nonlinearties. Electr. J. Differ. Equ. 2019(95), 1–12 (2019)

    MATH  Google Scholar 

  12. Keller, H.B., Cohen, D.S.: Some positive problems suggested by nonlinear heat generators. J. Math. Mech. 16(12), 1361–1376 (1967)

    MathSciNet  MATH  Google Scholar 

  13. Kovaxik, O., Rakosnik, J.: On spaces \(L^{p(x)}\) and \(W^{k, p(x)}\). Czechoslov. Math. J. 41, 592–618 (1991)

    Article  MATH  Google Scholar 

  14. Kim, Y.H., Wang, L., Zhang, C.: Global bifurcation for a class of degenerate elliptic equations with variable exponents. J. Math. Anal. Appl. 371(2), 624–637 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ladyzhenskaya, A.O., Uraltseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)

    Google Scholar 

  16. Lazer, A.C., Mckenna, P.J.: On a singular nonlinear elliptic boundary value problem. Proc. Am. Math. Soc. 111(3), 721–730 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. Levine, Y., Rao, M.S.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66(4), 1383–1406 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lions, J.L.: Quelques methodes de resolution des problemes aux limites non-linearies. Dunod, GauthierVillars, Paris (1969)

    MATH  Google Scholar 

  19. Maso, G.D., Murat, F., Orsina, L., Prignet, A.: Renormalized solutions of elliptic equations with general measure data. Ann. Sc. Norm. Super. Pisa Cl. Sci. 28(4), 741–808 (1999)

    MathSciNet  MATH  Google Scholar 

  20. Rodrigues, J.F., Sanchon, M., Urbano, J.M.: The obstacle problem for nonlinear elliptic equations with variable growth and \(L^1\)-data. Monatsh. Math. 154, 303–322 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ruzicka, M.: Electro-Rheological Fluids: Modeling and Mathematical Theory. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  22. Vazquez, J.L.: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12(1), 191–202 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, C.: Entropy solitions for nonlinear elliptic with variable exponents. Electr. J. Differ. Equ 2014(92), 1–14 (2014)

    Google Scholar 

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Acknowledgements

We are very grateful to the anonymous referees for their valuable suggestions that improved the article.

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Authors and Affiliations

  1. Labo DGTIC, Department of Mathematics, CRMEF Casablanca-Settat, El Jadida, Morocco

    J. Igbida

  2. Labo Math Appli, Faculty of Sciences, B. P. 20, El Jadida, Morocco

    N. Elharrar & H. Talibi

Authors
  1. J. Igbida
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  2. N. Elharrar
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  3. H. Talibi
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Correspondence to N. Elharrar.

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Igbida, J., Elharrar, N. & Talibi, H. Weighted \(p(\cdot )\)-Laplacian problem with nonlinear singular terms. Ricerche mat 72, 45–62 (2023). https://doi.org/10.1007/s11587-021-00590-4

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