The equation $-\Delta 𝑢-\lambda {\displaystyle \frac{𝑢}{{\left|𝑥\right|}^{\mathbf{2}}}}={\left|\nabla 𝑢\right|}^{𝑝}+𝑐𝑓\left(𝑥\right)$: The optimal power

Abdellaoui, Boumediene ; Peral, Ireneo

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 1, pp. 159-183

Résumé

We will consider the following problem

$$-\Delta u-\lambda \frac{u}{{\left|x\right|}^2}={\left|\nabla u\right|}^p+cf,u>0\text{in}\Omega ,$$

where $\Omega \subset {ℝ}^N$ is a domain such that $0\in \Omega$, $N\ge 3$, $c>0$ and $\lambda >0$. The main objective of this note is to study the precise threshold $p_{+}=p_{+}\left(\lambda \right)$ for which there is no very weak supersolution if $p\ge p_{+}\left(\lambda \right)$. The optimality of $p_{+}\left(\lambda \right)$ is also proved by showing the solvability of the Dirichlet problem when $1\le p<p_{+}\left(\lambda \right)$, for $c>0$ small enough and $f\ge 0$ under some hypotheses that we will prescribe.

Détail

Classification : 35D05, 35J10, 35J60, 46E30