AIMS Mathematics, 2020, 5(3): 1779-1798. doi: 10.3934/math.2020120

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Elliptic problems with singular nonlinearities of indefinite sign

Facultad de Matematica, Astronomia y Fisica, Universidad Nacional de Cordoba, Ciudad Universitaria, 5000 Cordoba, Argentina

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with $C^{1,1}$ boundary. We consider problems of the form $-\Delta u=\chi_{\left\{ u>0\right\}}\left( au^{-\alpha}-g\left( .,u\right) \right) $ in $\Omega,$ $u=0$ on $\partial\Omega,$ $u\geq0$ in $\Omega,$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $0\not \equiv a\in L^{\infty}\left( \Omega\right) ,$ $\alpha\in\left( 0,1\right) ,$ and $g:\Omega\times\left[ 0,\infty\right) \rightarrow\mathbb{R}$ is a nonnegative Carathéodory function. We prove, under suitable assumptions on $a$ and $g,$ the existence of nontrivial and nonnegative weak solutions $u\in H_{0}^{1}\left( \Omega\right) \cap L^{\infty}\left( \Omega\right) $ of the stated problem. Under additional assumptions, the positivity, $a.e.$ in $\Omega,$ of the found solution $u$, is also proved.
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1. H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011.

2. H. Brezis and X Cabre, Some simple nonlinear PDE's without solutions, Bollettino Della Unione Matematica Italiana, 2 (1998), 223-262.

3. A. Callegari and A. Nashman, A nonlinear singular boundary-value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math., 38 (1980), 275-281.    

4. M. M. Coclite and G. Palmieri, On a singular nonlinear Dirichlet problem, Comm. Part. Differ. Equat., 14 (1989), 1315-1327.    

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

6. M. G. Crandall, P. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Part. Differ. Equations, 2 (1977), 193-222.    

7. F. Cîrstea, M. Ghergu and V. Rădulescu, Combined effects of asymptotically linear and singular nonlinearities in bifurcation problems of Lane-Emden-Fowler type, J. Math. Pures Appl., 84 (2005), 493-508.    

8. Y. Chu, Y. Gao and W. Gao, Existence of solutions to a class of semilinear elliptic problem with nonlinear singular terms and variable exponent, J. Funct. Spaces, 2016 (2016), 1-11.

9. J. Dávila and M. Montenegro, Positive versus free boundary solutions to a singular elliptic equation, J. Anal. Math., 90 (2003), 303-335.    

10. J. Dávila and M. Montenegro, Existence and asymptotic behavior for a singular parabolic equation, Trans. Amer. Math. Soc., 357 (2005), 1801-1828    

11. M. A. del Pino, A global estimate for the gradient in a singular elliptic boundary value problem, Proc. R. Soc. Edinburgh Sect. A, 122 (1992), 341-352.    

12. J. I. Díaz and J. Hernández, Positive and free boundary solutions to singular nonlinear elliptic problems with absorption; An overview and open problems, Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems, Electron. J. Differ. Eq., Conf., 21 (2014), 31-44.

13. J. I. Díaz, J. M. Morel and L. Oswald, An elliptic equation with singular nonlinearity, Comm. Part. Diff. Eq., 12 (1987), 1333-1344.    

14. L. Dupaigne, M. Ghergu and V. Rădulescu, Lane-Emden-Fowler equations with convection and singular potential, J. Math. Pures Appl., 87 (2007), 563-581.    

15. D. G. De Figueiredo, Positive solutions of semilinear elliptic equations, in: Guedes de Figueiredo D., Hönig C.S. (eds) Differential Equations. Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 957 (1982), 34-87

16. W. Fulks and J. S. Maybee, A singular nonlinear equation, Osaka Math. J., 12 (1960), 1-19.

17. L. Gasiński and N. S. Papageorgiou, Nonlinear Elliptic Equations with Singular Terms and Combined Nonlinearities, Ann. Henri Poincaré, 13 (2012), 481-512.

18. M. Ghergu and V. D. Rădulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, 2008.

19. M. Ghergu and V. D. Rădulescu, Multi-parameter bifurcation and asymptotics for the singular Lane-Emden-Fowler equation with a convection term, Proc. Royal Soc. Edinburgh, Sect. A, 135 (2005), 61-84.    

20. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

21. T. Godoy and A. Guerin, Existence of nonnegative solutions for singular elliptic problems, Electron. J. Differential Equations, 191 (2016), 1-16.

22. T. Godoy and A. Guerin, Nonnegative solutions to some singular semilinear elliptic problems, Journal of Nonlinear Functional Analysis, 2017 (2017), 1-23.

23. T. Godoy and A. Guerin, Positive weak solutions of elliptic Dirichlet problems with singularities in both the dependent and the independent variables, Electron. J. Qual. Theory Differ. Equ., 54 (2019), 1-17.

24. T. Godoy and A. Guerin, Regularity of the lower positive branch for singular elliptic bifurcation problems, Electron. J. Differential Equations, 49 (2019), 1-32.

25. T. Godoy and A. Guerin, Existence of nonnegative solutions to singular elliptic problems, a variational approach, Discrete Contin. Dyn. Syst., Series A, 38 (2018), 1505-1525.    

26. A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730.    

27. M. Montenegro and A. Ponce, The sub-supersolution method for weak solutions, Proc. Amer. Math. Soc., 136 (2008), 2429-2438.    

28. N. H. Loc and K. Schmitt, Boundary value problems for singular elliptic equations, Rocky Mt. J. Math., 41 (2011), 555-572.    

29. N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Nonlinear Analysis-Theory and Methods, Springer Monographs in Mathematics, Springer, Cham, 2019.

30. N. S. Papageorgiou and G. Smyrlis, Nonlinear elliptic equations with singular reaction, Osaka J. Math., 53 (2016), 489-514.

31. A. C. Ponce, Selected problems on elliptic equations involving measures, arXiv:1204.0668 [v3], 2017.

32. V. D. Rădulescu, Singular phenomena in nonlinear elliptic problems. From blow-up boundary solutions to equations with singular nonlinearities, in: Handbook of Differential Equations: Stationary Partial Differential Equations, (M. Chipot, Editor), North-Holland Elsevier Science, Amsterdam, 4 (2007), 483-591.

33. W. Rudin, Functional analysis, International Series in Pure and Applied Mathematics, McGrawHill, Inc., New York, 1991.

34. J. C. Sabina de Lis, Hopf maximum principle revisited, Electron. J. Differential Equations, 115 (2015), 1-9.

35. J. Shi and M. Yao, On a singular nonlinear semilinear elliptic problem, Proc. R. Soc. Edinburgh, Sect A, 128 (1998), 1389-1401.    

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