The equation $-\Delta 𝑢-\lambda \frac{𝑢}{{|𝑥|}^{\mathbf{2}}}={|\nabla 𝑢|}^{𝑝}+𝑐𝑓\left(𝑥\right)$: The optimal power
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 1, pp. 159-183.

We will consider the following problem

 $-\Delta u-\lambda \frac{u}{{|x|}^{2}}={|\nabla u|}^{p}+c\phantom{\rule{0.166667em}{0ex}}f,\phantom{\rule{1em}{0ex}}u>0\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\Omega ,\phantom{\rule{1em}{0ex}}$
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.

Classification: 35D05,  35J10,  35J60,  46E30
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author = {Abdellaoui, Boumediene and Peral, Ireneo},
title = {The equation $-\Delta \textit {u}-\lambda \dfrac{\textit {u}}{|\textit {x}|^{\bf 2}}=|\nabla \textit {u}|^{\textit {p}}+ \textit {c} \textit {f}(\textit {x})$: {The} optimal power},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {159--183},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 6},
number = {1},
year = {2007},
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Abdellaoui, Boumediene; Peral, Ireneo. The equation $-\Delta \textit {u}-\lambda \dfrac{\textit {u}}{|\textit {x}|^{\bf 2}}=|\nabla \textit {u}|^{\textit {p}}+ \textit {c} \textit {f}(\textit {x})$: The optimal power. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 1, pp. 159-183. http://archive.numdam.org/item/ASNSP_2007_5_6_1_159_0/

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