Abstract
In this paper, we establish the existence of a positive solution to
where \(\Omega \) is a smooth bounded domain in \(\mathbb {R}^{n},~n\ge 1.\) Under certain conditions on \(k,f~\text {and}~h,\) using viscosity sub- and super solution method with the aid of comparison principle, we establish the existence of a unique positive viscosity solution. This work extends and complements the earlier works on semilinear and singular elliptic equations with sublinear nonlinearity.
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Bertozzi, A.L., Pugh, M.C.: Long-wave instabilities and saturation in thin film equations. Commun. Pure Appl. Math. 51(6), 625–661 (1998)
Bertozzi, A.L., Pugh, M.C.: Finite-time blow-up of solutions of some long-wave unstable thin film equations. Indiana Univ. Math. J. 49, 1323–1366 (2000)
Birindelli, I., Demengel, F.: Eigenvalue, maximum principle and regularity for fully nonlinear homogeneous operators. Commun. Pure Appl. Anal. 6(2), 335–366 (2007)
Birindelli, I., Demengel, F.: The Dirichlet problem for singular fully nonlinear operators. In: Discrete and Continuous Dynamical Systems, Dynamical systems and differential equations. Proceedings of the 6th AIMS International Conference, suppl, pp. 110–121 (2007)
Boccardo, L., Orsina, L.: Semilinear elliptic equations with singular nonlinearities. Calc. Var. PDEs 37, 363–380 (2010)
Boccardo, L., Orsina, L.: Sublinear elliptic equations with singular potentials. Adv. Nonlinear Stud. 12(2), 187–198 (2012)
Busca, J., Esteban, M.J., Quaas, A.: Nonlinear eigenvalues and bifurcation problems for Pucci’s operator. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(2), 187–206 (2005)
Cabré, X., Caffarelli, L.A.: Fully Nonlinear Elliptic Equation, American Mathematical Society Colloquium Publications, vol. 43. American Mathematical Society, Providence (1995)
Chhetri, M., Robinson, S.: Existence and multiplicity of positive solutions for classes of singular elliptic PDEs. J. Math. Anal. Appl. 357(1), 176–182 (2009)
Coclite, M.M., Palmieri, G.: On a singular nonlinear Dirichlet problem. Commun. Partial Differ. Equ. 14(10), 1315–1327 (1989)
Coclite, M.M.: On a singular nonlinear Dirichlet problem III. Nonlinear anal. 21, 547–564 (1993)
Crandall, M.G., Rabinowitz, P.H., Tartar, L.: On a Dirichlet problem with a singular nonlinearity. Commun. Part. Differ. Equ. 2(2), 193–222 (1977)
Crandall, M.G., Caffarelli, L., Kocan, M., Świech, A.: On viscosity solutions of fully nonlinear equations with measurable ingredients. Commun. Pure Appl. Math. 49(4), 365–398 (1996)
Dolcetta, I.C., Vitolo, A.: Gradient and Hölder estimates for positive solutions of Pucci type equations. C. R. Math. Acad. Sci. Paris 346(9–10), 527–532 (2008)
Dolcetta, I.C., Vitolo, A.: \(C^{1,\alpha }\) and Glaeser type estimates. Rend. Mat. Appl. 7(29), 17–27 (2009)
Felmer, P., Quaas, A., Sirakov, B.: Existence and regularity results for fully nonlinear equations with singularities. Math. Ann. 354(1), 377–400 (2012)
Felmer, P., Quaas, A., Sirakov, B.: Landesman–Lazer type results for second order Hamilton–Jacobi–Bellman equations. J. Func. Anal. 258(12), 4154–4182 (2010)
Godoy, T., Kaufmann, U.: On Dirichlet problems with singular nonlinearity of indefinite sign. J. Math. Anal. Appl. 428(2), 1239–1251 (2015)
Lazer, A.C., McKenna, P.J.: On a singular nonlinear elliptic boundary-value problem. Proc. Am. Math. Soc. 111, 721–730 (1991)
Papageorgiou, N.S., Rădulescu, V.: Combined effects of singular and sublinear nonlinearities in some elliptic problems. Nonlinear Anal. 109, 236–244 (2014)
Pelesko, J.A.: Mathematical modeling of electrostatic MEMS with tailored dielectric properties. SIAM J. Appl. Math. 62(3), 888–908 (2002)
Quaas, A., Sirakov, B.: Existence results for nonproper elliptic equations involving the Pucci operator. Commun. Part. Differ. Equ. 31(7), 987–1003 (2006)
Quaas, A., Sirakov, B.: Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators. Adv. Math. 218, 105–135 (2008)
Świech, A.: \(W^{1, p}\), Interior estimates for solutions of fully nonlinear, uniformly elliptic equations. Adv. Differ. Equ. 2(6), 1005–1027 (1997)
Winter, N.: \(W^{1, p}\) and \(W^{2, p}\)-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations. Z. Anal. Anwend. 28(2), 129–164 (2009)
Tyagi, J., Verma, R.B.: A survey on the existence, uniqueness and regularity questions to fully nonlinear elliptic partial differential equations. Differ. Equ. Appl. 8(2), 135–205 (2016)
Tyagi, J.: An existence of positive solutions to singular elliptic equations. Boll. Unione Mat. Ital. 7(1), 45–53 (2014)
Yijing, S., Duanzhi, Z.: The role of the power 3 for elliptic equations with negative exponents. Calc. Var. Partial Differ. Equ. 49, 909–922 (2014)
Yijing, S., Wu, S.: An exact estimate result for a class of singular equations with critical exponents. J. Func. Anal. 260(5), 1257–1284 (2011)
Yijing, S.: Compatibility phenomena in singular problems. Proc. R. Soc. Edinb. Sect. A 143(6), 1321–1330 (2013)
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Tyagi, J., Verma, R.B. Positive Solution of Extremal Pucci’s Equations with Singular and Sublinear Nonlinearity. Mediterr. J. Math. 14, 148 (2017). https://doi.org/10.1007/s00009-017-0950-6
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DOI: https://doi.org/10.1007/s00009-017-0950-6