Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 1, p. 117-158
We investigate the following quasilinear and singular problem, to2.7cm-Δ p u=λ u δ +u q inΩ;u| Ω =0,u>0inΩ,to2.7cm(P) where Ω is an open bounded domain with smooth boundary, 1<p<, p-1<qp * -1, λ>0, and 0<δ<1. As usual, p * =Np N-p if 1<p<N, p * (p,) is arbitrarily large if p=N, and p * = if p>N. We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in W 0 1,p (Ω). While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle and a regularity result for solutions to problem (P) in C 1,β (Ω ¯) with some β(0,1). Furthermore, we show that δ<1 is a reasonable sufficient (and likely optimal) condition to obtain solutions of problem (P) in C 1 (Ω ¯).
Classification:  35J65,  35J20,  35J70
@article{ASNSP_2007_5_6_1_117_0,
     author = {Giacomoni, Jacques and Schindler, Ian and Tak\'a\v c, Peter},
     title = {Sobolev versus H\"older local minimizers and existence of multiple solutions for a singular quasilinear equation},
     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 = {117-158},
     zbl = {1181.35116},
     mrnumber = {2341518},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2007_5_6_1_117_0}
}
Giacomoni, Jacques; Schindler, Ian; Takáč, Peter. Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 1, pp. 117-158. http://www.numdam.org/item/ASNSP_2007_5_6_1_117_0/

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