# American Institute of Mathematical Sciences

November  2013, 33(11&12): 4923-4944. doi: 10.3934/dcds.2013.33.4923

## On a Dirichlet problem in bounded domains with singular nonlinearity

 1 Dipartimento di Matematica, Università di Bari, via Orabona 4, 70125 Bari, Italy 2 Department of Mathematics, University of Bari, Via E. Orabona 4, 70125 Bari, Italy

Received  May 2012 Published  May 2013

In this paper we prove the existence and regularity of positive solutions of the homogeneous Dirichlet problem \begin{equation*} -Δ u=g(x,u)     in     \Omega,         u=0    on     ∂ \Omega, \end{equation*} where $g(x,u)$ can be singular as $u\rightarrow0^+$ and $0\le g(x,u)\le\frac{\varphi_0(x)}{u^p}$ or $0\le$ $g(x,u)$ $\le$ $\varphi_0(x)(1+\frac{1}{u^p})$, with $\varphi_0 \in L^m(\Omega), 1 ≤ m.$ There are no assumptions on the monotonicity of $g(x,\cdot)$ and the existence of super- or sub-solutions.
Citation: Giuseppe Maria Coclite, Mario Michele Coclite. On a Dirichlet problem in bounded domains with singular nonlinearity. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 4923-4944. doi: 10.3934/dcds.2013.33.4923
##### References:
 [1] L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var. Partial Differential Equations, 37 (2010), 363-380. doi: 10.1007/s00526-009-0266-x.  Google Scholar [2] H. Brezis and X. Cabré, Some simple nonlinear PDE's without solutions, Boll. Un. Mat. Ital., 1 (1998), 223-262.  Google Scholar [3] M. M. Coclite, On a singular nonlinear dirichlet problem - II, Boll. Un. Mat. Ital., 5 (1991), 955-975.  Google Scholar [4] M. M. Coclite, On a singular nonlinear Dirichlet problem - III, Nonlinear Anal., 21 (1993), 547-564. doi: 10.1016/0362-546X(93)90010-P.  Google Scholar [5] G. M. Coclite and M. M. Coclite, Elliptic perturbations for Hammerstein equations with singular nonlinear term, Electron. J. Diff. Eqns., 2006 (2006), 23 pp. (electronic).  Google Scholar [6] M. G. Crandal, P. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 2 (1977), 193-222. doi: 10.1080/03605307708820029.  Google Scholar [7] G. Stampacchia, Le problème de Dirichlet poue les équations elliptiques du second order à coefficientes discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258. doi: 10.5802/aif.204.  Google Scholar [8] Z. Zhao, Green function for Schrödinger operator and conditioned Feynman-Kac gauge, J. Math. Anal. Appl., 116 (1986), 309-334. doi: 10.1016/S0022-247X(86)80001-4.  Google Scholar

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##### References:
 [1] L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var. Partial Differential Equations, 37 (2010), 363-380. doi: 10.1007/s00526-009-0266-x.  Google Scholar [2] H. Brezis and X. Cabré, Some simple nonlinear PDE's without solutions, Boll. Un. Mat. Ital., 1 (1998), 223-262.  Google Scholar [3] M. M. Coclite, On a singular nonlinear dirichlet problem - II, Boll. Un. Mat. Ital., 5 (1991), 955-975.  Google Scholar [4] M. M. Coclite, On a singular nonlinear Dirichlet problem - III, Nonlinear Anal., 21 (1993), 547-564. doi: 10.1016/0362-546X(93)90010-P.  Google Scholar [5] G. M. Coclite and M. M. Coclite, Elliptic perturbations for Hammerstein equations with singular nonlinear term, Electron. J. Diff. Eqns., 2006 (2006), 23 pp. (electronic).  Google Scholar [6] M. G. Crandal, P. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 2 (1977), 193-222. doi: 10.1080/03605307708820029.  Google Scholar [7] G. Stampacchia, Le problème de Dirichlet poue les équations elliptiques du second order à coefficientes discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258. doi: 10.5802/aif.204.  Google Scholar [8] Z. Zhao, Green function for Schrödinger operator and conditioned Feynman-Kac gauge, J. Math. Anal. Appl., 116 (1986), 309-334. doi: 10.1016/S0022-247X(86)80001-4.  Google Scholar
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