Semilinear problems with right-hand sides singular at $u=0$ = 0 which change sign

Casado-Díaz, Juan¹ ; Murat, François²

¹ a Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Spain
² b Laboratoire Jacques-Louis Lions, Sorbonne Université & CNRS, France

Annales de l'I.H.P. Analyse non linéaire, mai – juin 2021, Tome 38 (2021) no. 3, pp. 877-909.

Résumé

The present paper is devoted to the study of the existence of a solution u for a quasilinear second order differential equation with homogeneous Dirichlet conditions, where the right-hand side tends to infinity at u=0. The problem has been considered by several authors since the 70's. Mainly, nonnegative right-hand sides were considered and thus only nonnegative solutions were possible. Here we consider the case where the right-hand side can change sign but is non negative (finite or infinite) at u=0, while no restriction on its growth at u=0 is assumed on its positive part. We show that there exists a nonnegative solution in a sense introduced in the paper; moreover, this solution is stable with respect to the right-hand side and is unique if the right-hand side is nonincreasing in u. We also show that if the right-hand side goes to infinity at zero faster than 1/|u|, then only nonnegative solutions are possible. We finally prove by means of the study of a one-dimensional example that nonnegative solutions and even many solutions which change sign can exist if the growth of the right-hand side is 1/|u|γ with 0<γ<1.

Détail

DOI : 10.1016/j.anihpc.2020.09.001

Classification : 35J25
Keywords: Singular equations, Monotone operators, Existence, Uniqueness, Positive and nonpositive solutions

Bibliographie

[1] Boccardo, L.; Casado-Díaz, J. Some properties of solutions of some semilinear elliptic singular problems and applications to the $G$-convergence, Asymptot. Anal., Volume 86 (2014), pp. 1-15 | MR | Zbl

[2] Boccardo, L.; Gallouët, T.; Murat, F. Unicité de la solution de certaines équations elliptiques non linéaires, C. R. Acad. Sci. Paris, Ser. I, Volume 315 (1992), pp. 1159-1164 | MR | Zbl

[3] Boccardo, L.; Orsina, L. Semilinear elliptic equations with singular nonlinearities, Calc. Var. Partial Differ. Equ., Volume 37 (2010), pp. 363-380 | MR | Zbl | DOI

[4] Casado-Díaz, J.; Murat, F.; Porretta, A. Uniqueness results for pseudomonotone problems with p>2 , C. R. Acad. Sci. Paris, Ser. I, Volume 344 (2007), pp. 487-492 | MR | Zbl | DOI

[5] Chipot, M.; Michaille, G. Uniqueness results and monotonicity properties for strongly nonlinear elliptic variational inequalities, Ann. Sc. Norm. Super. Pisa, Volume 16 (1989), pp. 137-166 | MR | Zbl | mathdoc-id

[6] Cioranescu, D.; Murat, F.; Cioranescu, D.; Murat, F. Un terme étrange venu d'ailleurs, I et II (Brezis, H.; Lions, J.-L.; Cherkaev, A.; Kohn, R.V., eds.), Non-linear Partial Differential Equations and Their Applications. Collège de France Seminar, Vol. II and Vol. III, Pitman, London, 1982, pp. 98-138 (and 154–178 English translation:, A strange term coming from nowhere Topics in Mathematical Modeling of Composite Materials, 1997, Birkhäuser, Boston, pp. 44-93)

[7] Coclite, M.M.; Palmieri, G. On a singular nonlinear Dirichlet problem, Commun. Partial Differ. Equ., Volume 14 (1989), pp. 1315-1327 | MR | Zbl | DOI

[8] Crandall, M.G.; Rabinowitz, P.H.; Tartar, L. On a Dirichlet problem with a singular nonlinearity, Commun. Partial Differ. Equ., Volume 2 (1977), pp. 193-222 | MR | Zbl | DOI

[9] Dávila, J.; Montenegro, M. Positive versus free boundary solutions to a singular elliptic equation, J. Anal. Math., Volume 90 (2003), pp. 303-335 | MR | Zbl | DOI

[10] Diaz, J.I.; Morel, J.-M.; Oswald, L. An elliptic equation with singular nonlinearity, Commun. Partial Differ. Equ., Volume 12 (1987), pp. 1333-1344 | MR | Zbl | DOI

[11] A. Ferone, A. Mercaldo, S. Segura de León, A singular elliptic equation and a related functional, submitted. | MR

[12] Giachetti, D.; Martínez-Aparicio, P.J.; Murat, F. Advances in the study of singular semilinear elliptic problems (Ortegón Gallego, F.; Redondo Neble, M.V.; Rodríguez-Galván, J.R., eds.), Trends in Differential Equations and Applications, SEMA SIMAI Springer Ser., vol. 8, 2016, pp. 221-241 | MR | Zbl | DOI

[13] 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., Volume 107 (2017), pp. 41-77 | MR | Zbl | DOI

[14] 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. Super. Pisa, Volume 18 (2018), pp. 1395-1442 | MR | Zbl

[15] Giachetti, D.; Martínez-Aparicio, P.J.; Murat, F. Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at u=0 in a domain with many small holes, J. Funct. Anal., Volume 274 (2018), pp. 1747-1789 | MR | Zbl | DOI

[16] Giachetti, D.; Martínez-Aparicio, P.J.; Murat, F. On the definition of the solution to a semilinear elliptic problem with a strong singularity at u=0 , Nonlinear Anal., Theory Methods Appl., Volume 177 (2018), pp. 491-523 | MR | Zbl | DOI

[17] Leray, J.; Lions, J.-L. Quelques résultats de Visik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. Fr., Volume 93 (1965), pp. 97-107 | MR | Zbl | mathdoc-id | DOI

[18] Lions, J.-L. Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969 | MR | Zbl

[19] Stuart, C.A. Existence and approximation of solutions of non-linear elliptic equations, Math. Z., Volume 147 (1976), pp. 53-63 | MR | Zbl | DOI