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

### Juan Casado-Díaz

Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Spain### François Murat

Laboratoire Jacques-Louis Lions, Sorbonne Université & CNRS, France

## 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 \( 1/ |u|\right.^{\gamma }\right. \) with $0<γ<1$.

## Cite this article

Juan Casado-Díaz, François Murat, Semilinear problems with right-hand sides singular at u = 0 which change sign. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 3, pp. 877–909

DOI 10.1016/J.ANIHPC.2020.09.001