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Semilinear elliptic equations with singular nonlinearities. (English) Zbl 1187.35081

Summary: We prove existence, regularity and nonexistence results for problems whose model is \[ -\Delta u = \frac{f(x)}{u^{\gamma}}\quad {\text{in} \,\Omega}, \] with zero Dirichlet conditions on the boundary of an open, bounded subset \(\Omega \) of \({\mathbb{R}^{N}}\). Here, \(\gamma > 0\) and \(f\) is a nonnegative function on \(\Omega \). Our results depend on the summability of \(f\) in some Lebesgue spaces, and on the values of \(\gamma \) (which can be equal, larger or smaller than 1).

MSC:

35J61 Semilinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J67 Boundary values of solutions to elliptic equations and elliptic systems
35B25 Singular perturbations in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35A35 Theoretical approximation in context of PDEs
35B09 Positive solutions to PDEs
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[1] Alves C.O., Goncalves J.V., Maia L.: Singular nonlinear elliptic equations in \({\mathbb{R}^N}\) . Abstr. Appl. Anal. 3, 411–423 (1998) · Zbl 0965.35052
[2] Arcoya D., Carmona J., Leonori T., Martínez-Aparicio P., Orsina L., Petitta F.: Existence and nonexistence of solutions for singular quadratic quasilinear equations. J. Differential Equations 246, 4006–4042 (2009) · Zbl 1173.35051
[3] Boccardo L.: Dirichlet problems with singular and gradient quadratic lower order terms. ESAIM Control Optim. Calc. Var. 14, 411–426 (2008) · Zbl 1147.35034
[4] Boccardo L., Gallouët T.: Nonlinear elliptic equations with right hand side measures. Comm. Partial Differential Equations 17, 641–655 (1992) · Zbl 0812.35043
[5] Boccardo L., Gallouët T., Orsina L.: Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data. Ann. Inst. H. Poincaré Anal. Non Linéaire 13, 539–551 (1996) · Zbl 0857.35126
[6] Boccardo L., Orsina L.: Sublinear equations in L s . Houston J. Math. 20, 99–114 (1994) · Zbl 0803.35047
[7] Canino A., Degiovanni M.: A variational approach to a class of singular semilinear elliptic equations. J. Convex Anal. 11, 147–162 (2004) · Zbl 1073.35092
[8] Coclite M.M., Palmieri G.: On a singular nonlinear Dirichlet problem. Comm. Partial Differential Equations 14, 1315–1327 (1989) · Zbl 0692.35047
[9] Crandall M.G., Rabinowitz P.H., Tartar L.: On a Dirichlet problem with a singular nonlinearity. Comm. Partial Differential Equations 2, 193–222 (1977) · Zbl 0362.35031
[10] 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. 28, 741–808 (1999) · Zbl 0958.35045
[11] Hirano N., Saccon C., Shioji N.: Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities. Adv. Differential Equations 9, 197–220 (2004) · Zbl 1387.35287
[12] Lair A.V., Shaker A.W.: Entire solution of a singular semilinear elliptic problem. J. Math. Anal. Appl. 200, 498–505 (1996) · Zbl 0860.35030
[13] Lair A.V., Shaker A.W.: Classical and weak solutions of a singular semilinear elliptic problem. J. Math. Anal. Appl. 211, 371–385 (1997) · Zbl 0880.35043
[14] Lazer A.C., McKenna P.J.: On a singular nonlinear elliptic boundary-value problem. Proc. Amer. Math. Soc. 111, 721–730 (1991) · Zbl 0727.35057
[15] Martínez-Aparicio, P.: Singular Dirichlet problems with quadratic gradient (preprint) · Zbl 1206.35132
[16] Stampacchia G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15, 189–258 (1965) · Zbl 0151.15401
[17] Stuart C.A.: Existence and approximation of solutions of non-linear elliptic equations. Math. Z. 147, 53–63 (1976) · Zbl 0324.35037
[18] Zhang Z., Cheng J.: Existence and optimal estimates of solutions for singular nonlinear Dirichlet problems. Nonlinear Anal. 57, 473–484 (2004) · Zbl 1096.35050
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