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

Abdellaoui, Boumediene; Peral, Ireneo

The equation

- Δ 𝑢 - λ \frac{𝑢}{{| 𝑥 |}^{2}} = {| \nabla 𝑢 |}^{𝑝} + 𝑐 𝑓 (𝑥)

: 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

- Δ u - λ \frac{u}{{| x |}^{2}} = {| \nabla u |}^{p} + c f, u > 0 in Ω,

where

Ω \subset ℝ^{N}

is a domain such that

0 \in Ω

,

N \geq 3

,

c > 0

and

λ > 0

. The main objective of this note is to study the precise threshold

p_{+} = p_{+} (λ)

for which there is no very weak supersolution if

p \geq p_{+} (λ)

. The optimality of

p_{+} (λ)

is also proved by showing the solvability of the Dirichlet problem when

1 \leq p < p_{+} (λ)

, for

c > 0

small enough and

f \geq 0

under some hypotheses that we will prescribe.

MR Zbl | 1 citation dans Numdam

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},
     year = {2007},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 6},
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     language = {en},
     url = {https://www.numdam.org/item/ASNSP_2007_5_6_1_159_0/}
}

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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, Série 5, Tome 6 (2007) no. 1, pp. 159-183. https://www.numdam.org/item/ASNSP_2007_5_6_1_159_0/

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