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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 . 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 , while no restriction on its growth at 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 , 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 with .
Keywords: Singular equations, Monotone operators, Existence, Uniqueness, Positive and nonpositive solutions
Casado-Díaz, Juan 1 ; Murat, François 2
@article{AIHPC_2021__38_3_877_0,
author = {Casado-D{\'\i}az, Juan and Murat, Fran\c{c}ois},
title = {Semilinear problems with right-hand sides singular at $u = 0$ $ = 0 which change sign},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {877--909},
publisher = {Elsevier},
volume = {38},
number = {3},
year = {2021},
doi = {10.1016/j.anihpc.2020.09.001},
mrnumber = {4227055},
zbl = {1466.35111},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.09.001/}
}
TY - JOUR AU - Casado-Díaz, Juan AU - Murat, François TI - Semilinear problems with right-hand sides singular at $u = 0$ $ = 0 which change sign JO - Annales de l'I.H.P. Analyse non linéaire PY - 2021 SP - 877 EP - 909 VL - 38 IS - 3 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.09.001/ DO - 10.1016/j.anihpc.2020.09.001 LA - en ID - AIHPC_2021__38_3_877_0 ER -
%0 Journal Article %A Casado-Díaz, Juan %A Murat, François %T Semilinear problems with right-hand sides singular at $u = 0$ $ = 0 which change sign %J Annales de l'I.H.P. Analyse non linéaire %D 2021 %P 877-909 %V 38 %N 3 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.09.001/ %R 10.1016/j.anihpc.2020.09.001 %G en %F AIHPC_2021__38_3_877_0
Casado-Díaz, Juan; Murat, François. Semilinear problems with right-hand sides singular at $u = 0$ $ = 0 which change sign. Annales de l'I.H.P. Analyse non linéaire, mai – juin 2021, Tome 38 (2021) no. 3, pp. 877-909. doi: 10.1016/j.anihpc.2020.09.001
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