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Journal Articles Annales de l'Institut Henri Poincaré C, Analyse non linéaire Year : 2021

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

Juan Casado-Díaz
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François Murat
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Abstract

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. ------------------------ This paper has been published in Ann. Inst. H. Poincaré, Anal. non linéaire, 38, (2021), pp. 877-909, doi 10.1016/J.ANIHPC.2020.09.001
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hal-03878575 , version 1 (01-12-2022)

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Juan Casado-Díaz, François Murat. Semilinear problems with right-hand sides singular at u = 0 which change sign. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2021, 38 (3), pp.877-909. ⟨10.1016/J.ANIHPC.2020.09.001⟩. ⟨hal-03878575⟩
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