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, Serie 5, Volume 6 (2007) no. 1, pp. 117-158.

We investigate the following quasilinear and singular problem,

 $\text{t}o2.7cm\left\{\begin{array}{cc}-{\Delta }_{p}u=\frac{\lambda }{{u}^{\delta }}+{u}^{q}\phantom{\rule{1em}{0ex}}\hfill & \phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\Omega ;\hfill \\ {u|}_{\partial \Omega }=0,\phantom{\rule{1em}{0ex}}u>0\phantom{\rule{1em}{0ex}}\hfill & \phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\Omega ,\hfill \end{array}\right\\text{t}o2.7cm\phantom{\rule{4pt}{0ex}}\text{(P)}$
where $\Omega$ is an open bounded domain with smooth boundary, $1, $p-1, $\lambda >0$, and $0<\delta <1$. As usual, ${p}^{*}=\frac{Np}{N-p}$ if $1, ${p}^{*}\in \left(p,\infty \right)$ is arbitrarily large if $p=N$, and ${p}^{*}=\infty$ if $p>N$. We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in ${W}_{0}^{1,p}\left(\Omega \right)$. 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 ${C}^{1,\beta }\left(\overline{\Omega }\right)$ with some $\beta \in \left(0,1\right)$. Furthermore, we show that $\delta <1$ is a reasonable sufficient (and likely optimal) condition to obtain solutions of problem (P) in ${C}^{1}\left(\overline{\Omega }\right)$.

Classification: 35J65,  35J20,  35J70
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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},
pages = {117--158},
publisher = {Scuola Normale Superiore, Pisa},
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number = {1},
year = {2007},
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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, Serie 5, Volume 6 (2007) no. 1, pp. 117-158. http://archive.numdam.org/item/ASNSP_2007_5_6_1_117_0/

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