On singular elliptic equations with measure sources

Oliva, Francescantonio; Petitta, Francesco

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 1, pp. 289-308

We prove existence of solutions for a class of singular elliptic problems with a general measure as source term whose model is

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\Delta u=\frac{f\left(x\right)}{u^{\gamma }}+\mu \hfill & \text{in}\Omega ,\hfill \\ u=0\hfill & \text{on}\partial \Omega ,\hfill \\ u>0\hfill & \text{on}\Omega ,\hfill \end{array}\right.\end{array}$$

where $\Omega$ is an open bounded subset of ${ℝ}^N$. Here $\gamma >0$, $f$ is a nonnegative function on $\Omega$, and $\mu$ is a nonnegative bounded Radon measure on $\Omega$.

Classification : 35J60, 35J61, 35J75, 35R06
Keywords: Nonlinear elliptic equations, singular elliptic equations, measure data

D. Arcoya, J. Carmona, T. Leonori, P.J. Martínez-Aparicio, L. Orsina and F. Petitta, Existence and nonexistence of solutions for singular quadratic quasilinear equations. J. Differ. Equ. 246 (2009) 4006–4042. | Zbl | MR | DOI

D. Arcoya and L. Moreno-Mérida, Multiplicity of solutions for a Dirichlet problem with a strongly singular nonlinearity. Nonlinear Analysis 95 (2014) 281–291. | Zbl | MR | DOI

D. Arcoya, J. Carmona and P. J. Martínez-Aparicio, Bifurcation for quasilinear elliptic singular BVP. Commun. Partial Differ. Equ. 36 (2011) 670–692. | Zbl | MR | DOI

P. Benilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J.L. Vazquez, An $L^1$ theory of existence and uniqueness of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa 22 (1995) 240–273. | Zbl | MR | Numdam

L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms. ESAIM: COCV 14 (2008) 411–426. | Zbl | MR | Numdam

L. Boccardo and J. Casado-Díaz, Some properties of solutions of some semilinear elliptic singular problems and applications to the G-convergence. Asymptotic Analysis 86 (2014) 1–15. | Zbl | MR | DOI

L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87 (1989) 149–169. | Zbl | MR | DOI

L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Analysis 19 (1992) 581–597. | Zbl | MR | DOI

L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities. Calc. Var. Partial Differ. Equ. 37 (2010) 363–380. | Zbl | MR | DOI

L. Boccardo, F. Murat and J.P. Puel, Existence of bounded solutions for nonlinear unilateral problems. Ann. Mat. Pura Appl. 152 (1988) 183–196. | Zbl | MR | DOI

H. Brezis and X. Cabré, Some simple nonlinear PDE’s without solutions. Bollettino dell’Unione Matematica Italiana Serie 8 (1998) 223–262. | Zbl | MR

H. Brezis, M. Marcus and A.C. Ponce, Nonlinear elliptic equations with measures revisited. Vol. 163 of Ann. Math. Stud. Princeton University Press NJ (2007) 55–110. | Zbl | MR

A. Canino, Minimax methods for singular elliptic equations with an application to a jumping problem. J. Differ. Equ. 221 (2006) 210–223. | Zbl | MR | DOI

A. Canino and M. Degiovanni, A variational approach to a class of singular semilinear elliptic equations. J. Convex Analysis 11 (2004) 147–162. | Zbl | MR

A. Canino, M. Grandinetti and B. Sciunzi, Symmetry of solutions of some semilinear elliptic equations with singular nonlinearities. J. Differ. Equ. 255 (2013) 4437–4447. | Zbl | MR | DOI

G.M. Coclite and M.M. Coclite, On a Dirichlet problem in bounded domains with singular nonlinearity. Discrete Contin. Dyn. Syst. 33 (2013) 4923–4944. | Zbl | MR | DOI

M.G. Crandall, P.H. Rabinowitz and L. Tartar, On a dirichlet problem with a singular nonlinearity. Commun. Partial. Differ. Equ. 2 (1977) 193–222. | Zbl | MR | DOI

G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data. Annali della Scuola Normale Superiore di Pisa 28 (1999) 741–808. | Zbl | MR | Numdam

L.M. De Cave, Nonlinear elliptic equations with singular nonlinearities. Asymptotic Anal. 84 (2013) 181–195. | Zbl | MR | DOI

D. Giachetti, P. J. Martínez-Aparicio and F. Murat, Elliptic equations with mild singularities: existence and homogenization. Preprint (2015). | arXiv | MR

D. Giachetti, P. J. Martínez-Aparicio and F. Murat, Homogenization of singular semilinear elliptic equations in domains with small holes (preprint).

D. Giachetti, F. Petitta, S. Segura de Leon, A priori estimates for elliptic problems with a strongly singular gradient term and a general datum. Differ. Int. Equ. 26 (2013) 913–948. | Zbl | MR

A.C. Lazer and P.J. Mckenna, On a singular nonlinear elliptic boundary-value problem. Proc. Amer. Math. Soc. 111 (1991) 721–730. | Zbl | MR | DOI

J. Leray and J.L. Lions, Quelques résultats de Višik sur les problémes elliptiques semilinéaires par les méthodes de Minty et Browder. Bull. Soc. Math. France 93 (1965) 97–107. | Zbl | MR | Numdam | DOI

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

A. C. Ponce, Selected problems on elliptic equations involving measures. Preprint (2014). | arXiv

J. Serrin, Pathological solutions of elliptic differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18 (1964) 385–387. | Zbl | MR | Numdam

G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du seconde ordre à coefficientes discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189–258. | Zbl | MR | Numdam | DOI

Y. Sun and D. Zhang, The role of the power 3 for elliptic equations with negative exponents. Calc. Var. Partial Differ. Equ. 49 (2014) 909–922. | Zbl | MR | DOI