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Existence and Regularity Results for Some Elliptic Equations with Degenerate Coercivity and Singular Quadratic Lower-Order Terms

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Abstract

In this paper, we study the existence and regularity results for some elliptic equations with degenerate coercivity and singular quadratic lower-order terms with natural growth with respect to the gradient. The model problem is

$$\begin{aligned} \left\{ \begin{array}{ll} - \mathrm {div}\left( \frac{\nabla u}{(1+|u|)^{\gamma }}\right) +\frac{\vert \nabla u\vert ^{2}}{u^{\theta }}=f+u^{r} &{} \quad \text{ in }\ \Omega ,\\ u=0&{} \quad \text{ on }\ \partial \Omega , \end{array}\right. \end{aligned}$$
(0.1)

where \(\Omega \) is a bounded open subset in \(\mathbb {R}^{N}\), \(0<\theta <1\), \(\gamma >0\) and \(0<r<2-\theta \). We will prove existence results for solutions under various assumptions on the summability of the source f.

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References

  1. Arcoya, D., et al.: Existence and nonexistence of solutions for singular quadratic quasilinear equations. J. Differ. Equ. 246(10), 4006–4042 (2009)

    Article  MathSciNet  Google Scholar 

  2. Arcoya, D., de León, S.S.: Uniqueness of solutions for some elliptic equations with a quadratic gradient term. ESAIM Control Optim. Calculus Var. 16(2), 327–336 (2010)

    Article  MathSciNet  Google Scholar 

  3. Arcoya, D., et al.: Some elliptic problems with singular natural growth lower order terms. J. Differ. Equ. 11(249), 2771–2795 (2010)

    Article  MathSciNet  Google Scholar 

  4. Arcoya, D., Carmona, J., Martinez-Aparicio, P.J.: Bifurcation for quasilinear elliptic singular BVP. Commun. Partial Differ. Equ. 36(4), 670–692 (2011)

    Article  MathSciNet  Google Scholar 

  5. Alvino, A., et al.: Existence results for nonlinear elliptic equations with degenerate coercivity. Ann. Mat. Pura Appl. 182, 53–79 (2003)

    Article  MathSciNet  Google Scholar 

  6. Boccardo, L.: Dirichlet problems with singular and gradient quadratic lower order terms. ESAIM Control Optim. Calculus Var. 14, 411–426 (2008)

    Article  MathSciNet  Google Scholar 

  7. Boccardo, L.: A contribution to the theory of quasilinear elliptic equations and application to the minimization of integral functionals. Milan J. Math. 79(1), 193–206 (2011)

    Article  MathSciNet  Google Scholar 

  8. Boccardo, L., Brezis, H.: Some remarks on a class of elliptic equations with degenerate coercivity. Boll. Unione Mat. Ital. 6, 521–530 (2003)

    MathSciNet  MATH  Google Scholar 

  9. Boccardo, L., Dalli-Aglio, A., Orsina, L.: Existence and regularity results for some elliptic equations with degenerate coercivity. Atti Sem. Mat. Fis. Univ. Modena 46, 51–81 (1998)

    MathSciNet  MATH  Google Scholar 

  10. Boccardo, L., Gallouët, T.: Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87, 149–169 (1989)

    Article  MathSciNet  Google Scholar 

  11. Boccardo, L., Moreno-Mérida, L., Orsina, L.: A class of quasilinear Dirichlet problems with unbounded coefficients and singular quadratic lower order terms. Milan J. Math. 83, 157–176 (2015)

    Article  MathSciNet  Google Scholar 

  12. Boccardo, L., Murat, F.: Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal. 19, 581–597 (1992)

    Article  MathSciNet  Google Scholar 

  13. Boccardo, L., Murat, F., Puel, J.P.: \(L^{\infty }\)-estimate for some nonlinear elliptic partial differential equations and application to an existence result. SIAM J. Math. Anal. 23, 326–333 (1992)

    Article  MathSciNet  Google Scholar 

  14. Boccardo, L., Orsina, L., Porzio, M.M.: Existence results for quasilinear elliptic and parabolic problems with quadratic gradient terms and sources. Adv. Calculus Var. 4(4), 397–419 (2011)

    MathSciNet  MATH  Google Scholar 

  15. Chen, G.: Nonlinear elliptic equation with lower order term and degenerate coercivity. Math. Notes 93(1–2), 224–237 (2013)

    Article  MathSciNet  Google Scholar 

  16. Croce, G.: An elliptic problem with degenerate coercivity and a singular quadratic gradient lower order term. Discrete Contin. Dyn. Syst. 5(3), 507–530 (2012)

    Article  MathSciNet  Google Scholar 

  17. Croce, G.: The regularizing effects of some lower order terms in an elliptic equation with degenerate coercivity. Rend. Mat. 27, 299–314 (2007)

    MathSciNet  MATH  Google Scholar 

  18. Giachetti, D., Murat, F.: An elliptic problem with a lower order term having singular behaviour. Boll. Unione Mat. Ital. 2009, 2 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Giachetti, D., Petitta, F., De León, S.S.: Elliptic equations having a singular quadratic gradient term and a changing sign datum. Commun. Pure Appl. Anal. 11, 1875–1895 (2012)

    Article  MathSciNet  Google Scholar 

  20. Giachetti, D., Petitta, F., de León, S.S.: A priori estimates for elliptic problems with a strongly singular gradient term and a general datum. Differ. Integral Equ. 26(9/10), 913–948 (2013)

    MathSciNet  MATH  Google Scholar 

  21. Giovanni, P., Vitolo, A.: Problems for elliptic singular equations with a quadratic gradient term. J. Math. Anal. Appl. 334(1), 467–486 (2007)

    Article  MathSciNet  Google Scholar 

  22. Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires (1969)

  23. Martinez-Aparicio, P.J.: Singular Dirichlet problems with quadratic gradient. Boll. Unione Mat. Ital. 2, 559–574 (2009)

    MathSciNet  MATH  Google Scholar 

  24. Mercaldo, A., Peral, I.: Existence results for semilinear elliptic equations with some lack of coercivity. Proc. R. Soc. Edinb. Sect. A Math. 138(3), 569–595 (2008)

    Article  MathSciNet  Google Scholar 

  25. Oliva, F., Petitta, F.: On singular elliptic equations with measure sources. ESAIM Control Optim. Calculus Var. 22(1), 289–308 (2016)

    Article  MathSciNet  Google Scholar 

  26. Orsina, L.: Solvability of linear and semilinear eigenvalue problems with L1 data. Rend. Sem. Mat. Univ. Padova 90, 207–238 (1993)

    MathSciNet  MATH  Google Scholar 

  27. Porretta, A.: Uniqueness and homogeneization for a class of noncoercive operators in divergence form. Atti Sem. Mat. Fis. Univ. Modena 46(suppl), 915–936 (1998)

    MathSciNet  MATH  Google Scholar 

  28. Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15, 189–258 (1965)

    Article  MathSciNet  Google Scholar 

  29. Trudinger, N.S.: On Harnack type inequalities and their application to quasilinear elliptic equations. Commun. Pure Appl. Math. 20, 721–747 (1967)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculté des Sciences Économiques Commerciales et de Gestions, Université Yahia Fares, Pôle urbain, 26000, Médéa, Algeria

    Rezak Souilah

  2. Laboratoire d’EDP Non Linéaires et Histoires des Mathématiques ENS-Kouba, B.P. 92,  Vieux Kouba, 16050, Alger, Algeria

    Rezak Souilah

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Souilah, R. Existence and Regularity Results for Some Elliptic Equations with Degenerate Coercivity and Singular Quadratic Lower-Order Terms. Mediterr. J. Math. 16, 87 (2019). https://doi.org/10.1007/s00009-019-1360-8

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  • DOI: https://doi.org/10.1007/s00009-019-1360-8

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