Artículo
Semilinear problems with right-hand sides singular at u = 0 which change sign
Autor/es | Casado Díaz, Juan
![]() ![]() ![]() ![]() ![]() ![]() ![]() Murat, François |
Departamento | Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico |
Fecha de publicación | 2021-06-01 |
Fecha de depósito | 2022-11-11 |
Publicado en |
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Resumen | 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 = 0u=0. ... 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 = 0u=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 = 0u=0, while no restriction on its growth at u = 0u=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|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|\right.^{\gamma }\right. with 0 < \gamma < 10<γ<1. |
Cita | Casado Díaz, J. y Murat, F. (2021). Semilinear problems with right-hand sides singular at u = 0 which change sign. Annales de l'Institut Henri Poincaré. Analyse non linéaire, 38 (3), 877-909. https://doi.org/10.1016/J.ANIHPC.2020.09.001. |
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