In this paper we consider a semilinear elliptic equation with a strong singularity at $u=0$, namely \begin{equation*} \begin{cases} \dys u\geq 0 & \mbox{in } \Omega,\\ \displaystyle - div \,A(x) D u = F(x,u)& \mbox{in} \; \Omega,\\ u = 0 & \mbox{on} \; \partial \Omega,\\ \end{cases} \end{equation*} with $F(x,s)$ a Carath\'eodory function such that $$ 0\leq F(x,s)\leq \frac{h(x)}{\Gamma(s)}\,\,\mbox{ a.e. } x\in\Omega,\, \forall s>0, $$ with $h$ in some $L^r(\Omega)$ and $\Gamma$ a $C^1([0,+\infty[)$ function such that $\Gamma(0)=0$ and $\Gamma'(s)>0$ for every $s>0$. We introduce a notion of solution to this problem in the spirit of the solutions defined by transposition. This definition allows us to prove the existence and the stability of this solution, as well as its uniqueness when $F(x,s)$ is nonincreasing in $s$.

Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at u = 0 / Giachetti, D.; Martinez-Aparicio, P. J.; Murat, F.. - In: ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE. - ISSN 0391-173X. - STAMPA. - (2018). [10.2422/2036-2145.201612_008]

Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at u = 0

D. Giachetti;
2018

Abstract

In this paper we consider a semilinear elliptic equation with a strong singularity at $u=0$, namely \begin{equation*} \begin{cases} \dys u\geq 0 & \mbox{in } \Omega,\\ \displaystyle - div \,A(x) D u = F(x,u)& \mbox{in} \; \Omega,\\ u = 0 & \mbox{on} \; \partial \Omega,\\ \end{cases} \end{equation*} with $F(x,s)$ a Carath\'eodory function such that $$ 0\leq F(x,s)\leq \frac{h(x)}{\Gamma(s)}\,\,\mbox{ a.e. } x\in\Omega,\, \forall s>0, $$ with $h$ in some $L^r(\Omega)$ and $\Gamma$ a $C^1([0,+\infty[)$ function such that $\Gamma(0)=0$ and $\Gamma'(s)>0$ for every $s>0$. We introduce a notion of solution to this problem in the spirit of the solutions defined by transposition. This definition allows us to prove the existence and the stability of this solution, as well as its uniqueness when $F(x,s)$ is nonincreasing in $s$.
2018
Semilinear equations, singularity at $u=0$, existence, stability, uniqueness.
01 Pubblicazione su rivista::01a Articolo in rivista
Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at u = 0 / Giachetti, D.; Martinez-Aparicio, P. J.; Murat, F.. - In: ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE. - ISSN 0391-173X. - STAMPA. - (2018). [10.2422/2036-2145.201612_008]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1051334
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