Abstract
We study existence and uniqueness of solutions to a nonlinear elliptic boundary value problem with a general, and possibly singular, lower order term, whose model is
Here \(\Omega \) is an open bounded subset of \({\mathbb {R}}^N\) (\(N\ge 2\)), \(\Delta _p u:= {\text {div}}(|\nabla u|^{p-2}\nabla u)\) (\(1<p<N\)) is the p-laplacian operator, \(\mu \) is a nonnegative bounded Radon measure on \(\Omega \) and H(s) is a continuous, positive and finite function outside the origin which grows at most as \(s^{-\gamma }\), with \(\gamma \ge 0\), near zero.
Article PDF
Similar content being viewed by others
References
Anello, G., Faraci, F.: On a singular semilinear elliptic problem with an asymptotically linear nonlinearity. Proc. R. Soc. Edinb. Sect. A 146, 59–77 (2016)
Anello, G., Faraci, F.: Two solutions for an elliptic problem with two singular terms. Calc. Var. Partial Differ. Equ. 56, 56–91 (2017)
Arcoya, D., Moreno-Mérida, L.: Multiplicity of solutions for a Dirichlet problem with a strongly singular nonlinearity. Nonlinear Anal. 95, 281–291 (2014)
Benilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vazquez, J.L.: An \(L^1\) theory of existence and uniqueness of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa 22, 240–273 (1995)
Boccardo, L., Casado-Díaz, J.: Some properties of solutions of some semilinear elliptic singular problems and applications to the G-convergence. Asymptot. Anal. 86, 1–15 (2014)
Boccardo, L., Gallouët, T.: Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87, 149–169 (1989)
Boccardo, L., Orsina, L.: Semilinear elliptic equations with singular nonlinearities. Calc. Var. PDEs 37, 363–380 (2010)
Boccardo, L., Murat, F., Puel, J.P.: Existence of bounded solutions for nonlinear unilateral problems. Ann. Mat. Pura Appl. 152, 183–196 (1988)
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)
Bougherara, B., Giacomoni, J., Hernández, J.: Existence and regularity of weak solutions for singular elliptic problems. Electron. J. Differ. Equ. Conf. 22, 19–30 (2015)
Canino, A.: Minimax methods for singular elliptic equations with an application to a jumping problem. J. Differ. Equ. 221, 210–223 (2006)
Canino, A., Degiovanni, M.: A variational approach to a class of singular semilinear elliptic equations. J. Convex Anal. 11, 147–162 (2004)
Canino, A., Grandinetti, M., Sciunzi, B.: Symmetry of solutions of some semilinear elliptic equations with singular nonlinearities. J. Differ. Equ. 255, 4437–4447 (2013)
Canino, A., Sciunzi, B., Trombetta, A.: Existence and uniqueness for \(p\)-Laplace equations involving singular nonlinearities. NoDEA Nonlinear Differ. Equ. Appl. 23, 8 (2016)
Canino, A., Montoro, L., Sciunzi, B.: The moving plane method for singular semilinear elliptic problems. Nonlinear Anal. 156, 61–69 (2017)
Carmona, J., Martínez-Aparicio, P.J.: A singular semilinear elliptic equation with a variable exponent. Adv. Nonlinear Stud. 16, 491–498 (2016)
Coclite, G.M., Coclite, M.M.: On a Dirichlet problem in bounded domains with singular nonlinearity. Discrete Contin. Dyn. Syst. 33, 4923–4944 (2013)
Coclite, G.M., Coclite, M.M.: On the summability of weak solutions for a singular Dirichlet problem in bounded domains. Adv. Differ. Equ. 19, 585–612 (2014)
Crandall, M.G., Rabinowitz, P.H., Tartar, L.: On a dirichlet problem with a singular nonlinearity. Commun. Partial Differ. Equ. 2, 193–222 (1977)
Dal Maso, G., Murat, F., Orsina, L., Prignet, A.: Renormalized solutions of elliptic equations with general measure data. Ann. Sc. Norm. Super. Pisa Cl. Sci. 4, 741–808 (1999)
De Cave, L.M.: Nonlinear elliptic equations with singular nonlinearities. Asymptot. Anal. 84, 181–195 (2013)
De Cave, L.M., Oliva, F.: Elliptic equations with general singular lower order terms and measure data. Nonlinear Anal. 128, 391–411 (2015)
De Cicco, V., Giachetti, D., Oliva, F., Petitta, F.: Dirichlet problems for singular elliptic equations with general nonlinearities. arXiv:1801.03444
Dìaz, J.I., Hernndez, J., Rakotoson, J.M.: On very weak positive solutions to some semilinear elliptic problems with simultaneous singular nonlinear and spatial dependence terms. Milan J. Math. 79, 233–245 (2011)
El Berdan, N., Díaz, J.I., Rakotoson, J.M.: The uniform Hopf inequality for discontinuous coefficients and optimal regularity in BMO for singular problems. J. Math. Anal. Appl. 437, 350–379 (2016)
Fukushima, M., Sato, K., Taniguchi, S.: On the closable part of pre-Dirichlet forms and the fine support of the underlying measures. Osaka J. Math. 28, 517–535 (1991)
Giachetti, D., Martínez-Aparicio, P.J., Murat, F.: Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at \(u = 0\). Ann. Sc. Norm. Sup. Pisa. https://doi.org/10.2422/2036-2145.201612_008
Giachetti, D., Martínez-Aparicio, P.J., Murat, F.: A semilinear elliptic equation with a mild singularity at \(u = 0\): existence and homogenization. J. Math. Pures Appl. 107, 41–77 (2017)
Goncalves, J.V.A., Carvalho, M.L.M., Santos, A.: Existence and regularity of positive solutions of quasilinear elliptic problems with singular semilinear term. arXiv:1703.08608v1
Gui, C., Lin, F.: Regularity of an elliptic problem with a singular nonlinearity. Proc. R. Soc. Edinb. Sect. A 123, 1021–1029 (1993)
Kilpeläinen, T., Kinnunen, J., Martio, O.: Sobolev spaces with zero boundary values on metric spaces. Potential Anal. 12, 233–247 (2000)
Klimsiak, T.: Semilinear elliptic equations with Dirichlet operator and singular nonlinearities. J. Funct. Anal. 272, 929–975 (2017)
Lazer, A.C., McKenna, P.J.: On a singular nonlinear elliptic boundary-value problem. Proc. Am. Math. Soc. 111, 721–730 (1991)
Leray, J., Lions, J.L.: Quelques résulatats de Višik sur les problémes elliptiques nonlinéaires par les méthodes de Minty-Browder. Bull. Soc. Math. France 93, 97–107 (1965)
Murat, F., Porretta, A.: Stability properties, existence and nonexistence of renormalized solutions for elliptic equations with measure data. Commun. Partial Differ. Equ. 27, 2267–2310 (2002)
Oliva, F., Petitta, F.: On singular elliptic equations with measure sources. ESAIM Control Optim. Calc. Var. 22, 289–308 (2016)
Oliva, F., Petitta, F.: Finite and infinite energy solutions of singular elliptic problems: existence and uniqueness. J. Differ. Equ. 264(1), 311–340 (2018)
Orsina, L., Petitta, F.: A Lazer–McKenna type problem with measures. Differ. Integral Equ. 29, 19–36 (2016)
Ponce, A.: Elliptic PDEs Measures and Capacities. Tracts in Mathematics, vol. 23. European Mathematical Society, Europe (2015)
Stuart, C.A.: Existence and approximation of solutions of nonlinear elliptic equations. Math. Z. 147, 5363 (1976)
Sun, Y., Zhang, D.: The role of the power 3 for elliptic equations with negative exponents. Calc. Var. PDEs 49, 909–922 (2014)
Trombetta, A.: A note on symmetry of solutions for a class of singular semilinear elliptic problems. Adv. Nonlinear Stud. 16, 499–507 (2016)
Trudinger, N.S.: On Harnack type inequalities and their application to quasilinear elliptic equations. Commun. Pure Appl. Math. 20, 721–747 (1967)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
De Cave, L.M., Durastanti, R. & Oliva, F. Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data. Nonlinear Differ. Equ. Appl. 25, 18 (2018). https://doi.org/10.1007/s00030-018-0509-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00030-018-0509-7