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Regularizing effects concerning elliptic equations with a superlinear gradient term

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Abstract

We consider the homogeneous Dirichlet problem for an elliptic equation driven by a linear operator with discontinuous coefficients and having a subquadratic gradient term. This gradient term behaves as \(g(u)|\nabla u|^q\), where \(1<q<2\) and g(s) is a continuous function. Data belong to \(L^m(\Omega )\) with \(1\le m <\frac{N}{2}\) as well as measure data instead of \(L^1\)-data, so that unbounded solutions are expected. Our aim is, given \(1\le m<\frac{N}{2}\) and \(1<q<2\), to find the suitable behaviour of g close to infinity which leads to existence for our problem. We show that the presence of g has a regularizing effect in the existence and summability of the solution. Moreover, our results adjust with continuity with known results when either g(s) is constant or \(q=2\).

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References

  1. Alvino, A., Ferone, V., Mercaldo, A.: Sharp a priori estimates for a class of nonlinear elliptic equations with lower order terms. Ann. Mat. Pura Appl. (4) 194(4), 1169–1201 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vázquez, J.L.: An \(L^1\)-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22(2), 241–273 (1995)

    MathSciNet  Google Scholar 

  3. Bensoussan, A., Boccardo, L., Murat, F.: On a nonlinear partial differential equation having natural growth terms and unbounded solution. Ann. Inst. Henri Poincaré Anal. Non Linéaire 5(4), 347–364 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boccardo, L., Gallouët, T.: Nonlinear elliptic equations with right hand side measures. Commun. Partial Differ. Equ. 17, 189–258 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Boccardo, L., Gallouët, T., Orsina, L.: Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data. Ann. Inst. Henri Poincaré Anal. Non Linéaire 13(5), 539–551 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Boccardo, L., Murat, F., Puel, J.-P.: Existence de solutions non bornées pour certaines équations quasi-linéaires. Port. Math. 41, 507–534 (1982)

    MATH  Google Scholar 

  8. Boccardo, L., Murat, F., Puel, J–.P.: Résultats d’existence pour certains problèmes elliptiques quasilinéaires. Ann. Sc. Norm. Super. Pisa Cl. Sci. IV Ser. 11, 213–235 (1984)

    MATH  Google Scholar 

  9. Boccardo, L., Murat, F., Puel, J–.P.: Existence of bounded solutions for non linear elliptic unilateral problems. Ann. Mat. Pura Appl. IV Ser. 152, 183–196 (1988)

    Article  MATH  Google Scholar 

  10. 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(2), 326–333 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bottaro, G., Marina, M.E.: Problema di Dirichlet per equazioni ellittiche di tipo variazionale su insiemi non limitati. Boll. Unione Mat. Ital. IV Ser. 8, 46–56 (1973)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  13. Del Vecchio, T., Porzio, M.M.: Existence results for a class of non coercive Dirichlet problems. Ric. Mat. 44(2), 421–438 (1995)

    MathSciNet  MATH  Google Scholar 

  14. Ferone, V., Murat, F.: Quasilinear problems having quadratic growth in the gradient: an existence result when the source term is small. In: Équations aux dérivées partielles et applications, pp. 497–515. Gauthier-Villars, Éd. Sci. Méd. Elsevier, Paris (1998)

  15. Ferone, V., Murat, F.: Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small. Nonlinear Anal. Theory Methods Appl. Ser. A Theory Methods 42(7), 1309–1326 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ferone, V., Murat, F.: Nonlinear elliptic equations with natural growth in the gradient and source terms in Lorentz spaces. J. Differ. Equ. 256(2), 577–608 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Grenon, N., Murat, F., Porretta, A.: Existence and a priori estimate for elliptic problems with subquadratic gradient dependent terms. C. R. Math. Acad. Sci. Paris 342(1), 23–28 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Grenon, N., Murat, F., Porretta, A.: A priori estimates and existence for elliptic equations with gradient dependent terms. Ann. Sc. Norm. Super. Pisa Cl. Sci. 13(1), 137–205 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Leone, Ch., Porretta, A.: Entropy solutions for nonlinear elliptic equations in \(L^1\). Nonlinear Anal. Theory Methods Appl. 32(3), 325–334 (1998)

    Article  Google Scholar 

  20. Leray, J., Lions, J.-L.: Quelques résultats de Visik sur les problèmes elliptiques non linégreenes par les méthodes de Minty–Browder. Bull. Soc. Math. Fr. 93, 97–107 (1965)

    Article  MATH  Google Scholar 

  21. López-Martínez, S.: A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems. Adv. Nonlinear Anal. 9(1), 1351–1382 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  22. Murat, F.: Soluciones renormalizadas de EDP elipticas no lineales. Laboratoire d’Analyse Numérique de l’Université Paris VI, Technical report R93023 (1993)

  23. Porretta, A.: Some remarks on the regularity of solutions for a class of elliptic equations with measure data. Houst. J. Math. 26, 183–213 (2000)

    MathSciNet  MATH  Google Scholar 

  24. Porretta, A.: Nonlinear equations with natural growth terms and measure data. Electron. J. Differ. Equ. Conf. 09, 183–202 (2002)

    MathSciNet  MATH  Google Scholar 

  25. Porretta, A., Segura de León, S.: Nonlinear elliptic equations having a gradient term with natural growth. J. Math. Pures Appl. (9) 85(3), 465–492 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Segura de León, S.: Existence and uniqueness for \(L^1\) data of some elliptic equations with natural growth. Adv. Differ. Equ. 8(11), 1377–1408 (2003)

    MathSciNet  Google Scholar 

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Acknowledgements

The third author has been partially supported by the Spanish Ministerio de Ciencia, Innovación y Universidades and FEDER, under Project PGC2018–094775–B–I00.

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Correspondence to Martina Magliocca.

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Latorre, M., Magliocca, M. & Segura de León, S. Regularizing effects concerning elliptic equations with a superlinear gradient term. Rev Mat Complut 34, 297–356 (2021). https://doi.org/10.1007/s13163-020-00353-z

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