The equation -Δ𝑢-λ𝑢 |𝑥| 2 =|𝑢| 𝑝 +𝑐𝑓(𝑥): The optimal power
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 1, p. 159-183
We will consider the following problem -Δu-λu |x| 2 =|u| p +cf,u>0inΩ, where Ω N is a domain such that 0Ω, N3, 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 pp + (λ). The optimality of p + (λ) is also proved by showing the solvability of the Dirichlet problem when 1p<p + (λ), for c>0 small enough and f0 under some hypotheses that we will prescribe.
Classification:  35D05,  35J10,  35J60,  46E30
@article{ASNSP_2007_5_6_1_159_0,
     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},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 6},
     number = {1},
     year = {2007},
     pages = {159-183},
     zbl = {1181.35080},
     mrnumber = {2341519},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2007_5_6_1_159_0}
}
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. http://www.numdam.org/item/ASNSP_2007_5_6_1_159_0/

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