AIMS Mathematics, 2018, 3(1): 233-252. doi: 10.3934/Math.2018.1.233

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Multiple finite-energy positive weak solutions to singular elliptic problems with a parameter

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

Consider the problem $-\Delta u=a\left( x\right) u^{-\alpha}+f\left(\lambda,x,u\right) $ in $\Omega,$ $u=0$ on $\partial\Omega,$ $u>0$ in$\Omega,$ \ where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with $C^{2}$ boundary, $0\leqa\in L^{\infty}\left( \Omega\right) ,$ $0<\alpha<3 and="" f="" left="" lambda="" x="" right="" is="" nonnegative="" and="" superlinear="" with="" subcritical="" growth="" at="" infty="" we="" prove="" that="" if="" f="" satisfies="" some="" additional="" conditions="" then="" for="" some="" lambda="">0 ,$ there are at least two weak solutions in$H_{0}^{1}\left( \Omega\right) \cap C\left( \overline{\Omega}\right) $ if$\lambda\in\left( 0,\Lambda\right)$, and there is no weak solution in$H_{0}^{1}\left( \Omega\right) \cap L^{\infty}\left( \Omega\right) $ if$\lambda>\Lambda.$ We also prove that, for each $\lambda\in\left[0,\Lambda\right] $, there exists a unique minimal weak solution $u_{\lambda}$ in $H_{0}^{1}\left( \Omega\right) \cap L^{\infty}\left( \Omega\right) $, which is strictly increasing in $\lambda.$
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1. H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620–709.

2. C. Aranda and T. Godoy, Existence and multiplicity of positive solutions for a singular problem associated to the p-Laplacian operator, Electron. J. Differ. Eq., 132 (2004), 1–15.

3. B. Bougherara, J. Giacomoni and J. Hernández, Existence and regularity of weak solutions for singular elliptic problems, 2014 Madrid Conference on Applied Mathematics in honor of Alfonso Casal, Electron. J. Differ. Equ. Conf., 22 (2015), 19–30.

4. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.

5. H. Brezis, M. Marcus and A. C. Ponce, Nonlinear elliptic equations with measures revisited, In Mathematical Aspects of Nonlinear Dispersive Equations, (J. Bourgain, C. Kenig S. Klainerman Eds.), Annals of Mathematics Studies, 163, Princeton University Press, Princeton, NJ, 2007, 55–110.

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

7. T. Cazenave, An introduction to semilinear elliptic equations, Editora do IM-UFRJ, Rio de Janeiro, 2006.

8. M. Chhetri, P. Drábek, and R. Shivaji, Analysis of positive solutions for classes of quasilinear singular problems on exterior domains, Adv. Nonlinear Anal., 6 (2017), 447–459.

9. M. M. Coclite and G. Palmieri, On a singular nonlinear Dirichlet problem, Commun. Part. Diff. Eq., 14 (1989), 1315–1327.

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

11. M. G. Crandall, P. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Commun. Part. Diff. Eq., 2 (1977), 193–222.

12. 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.

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

14. D. G. De Figueiredo, Positive solutions of semilinear elliptic equations, Lect. Notes Math., Springer, 957 (1982), 34–87.

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

16. 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 (2012). Electron. J. Differ. Equ. Conf., 21 (2014), 31–44.

17. J. I. Diaz, J. M. Morel and L. Oswald, An elliptic equation with singular nonlinearity, Commun. Part. Diff. Eq., 12 (1987), 1333–1344.

18. Du, Y., Order structures and topological methods in nonlinear partial differential equations, Vol 1: Maximum Principles and Applications, World Scientific Publishing Co., Pte., Ltd., Singapore, 2006.

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

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

21. M. Ghergu and V. D. Rădulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, No 37, 2008.

22. M. Ghergu and V. D. Rădulescu, Bifurcation and asymptotics for the Lane-Emden-Fowler equation, C. R. Math. Acad. Sci. Paris, 337 (2003), 259–264.

23. M. Ghergu and V. D. Rădulescu, Sublinear singular elliptic problems with two parameters, J. Differ. Equations, 195 (2003), 520–536.

24. M. Ghergu and V. D. Rădulescu, Ground state solutions for the singular Lane-Emden-Fowler equation with sublinear convection term, J. Math. Anal. Appl., 333 (2007), 265–273.

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

26. J. Giacomoni, I. Schindler and P. Takac, Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation, Ann. Scuola Norm-Sci, 6 (2007), 117–158.

27. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin Heidelberg New York, 2001.

28. T. Godoy and A. Guerin, Existence of nonnegative solutions for singular elliptic problems, Electronic Journal of Dierential Equations, 2016 (2016), 1–16.

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

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

31. T. Godoy and A. Guerin, Multiplicity of weak solutions to subcritical singular elliptic Dirichlet problems, Electron. J. Qual. Theory Differ. Equ., 2017 (2017), No. 100, 1–30.

32. U. Kaufmann and I. Medri, One-dimensional singular problems involving the p-Laplacian and nonlinearities indefinite in sign, Adv. Nonlinear Anal., 5 (2016), 251–259.

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

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

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

36. F. Oliva and F. Petitta, Finite and infinite energy solutions of singular elliptic problems: Existence and uniqueness, J. Differ. Equations, 264 (2018), 311–340.

37. N. S. Papageorgiou and V. D. Rădulescu, Combined effects of singular and sublinear nonlinearities in some elliptic problems, Nonlinear Anal-Theor, 109 (2014), 236–244.

38. 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, Vol. 4 (M. Chipot, Editor), North-Holland Elsevier Science, Amsterdam, 2007, pp. 483–591.

39. K. Saoudi, P. Agarwal and M. Mursaleen, A multiplicity result for a singular problem with subcritical nonlinearities, Journal of Nonlinear Functional Analysis, 2017 (2017), Article ID 33, 1–18.

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

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