Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation

Giacomoni, Jacques; Schindler, Ian; Takáč, Peter

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 1, pp. 117-158.

We investigate the following quasilinear and singular problem,

$$\text{t}o2.7cm\left\{\begin{array}{cc}-{\Delta }_{p}u=\frac{\lambda }{u^{\delta }}+u^q\hfill & \text{in}\Omega ;\hfill \\ {u|}_{\partial \Omega }=0,u>0\hfill & \text{in}\Omega ,\hfill \end{array}\right.\text{t}o2.7cm\text{(P)}$$

where $\Omega$ is an open bounded domain with smooth boundary, $1<p<\infty$, $p-1<q\le p^{*}-1$, $\lambda >0$, and $0<\delta <1$. As usual, $p^{*}=\frac{Np}{N-p}$ if $1<p<N$, $p^{*}\in (p,\infty )$ 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 $\beta \in (0,1)$. Furthermore, we show that $\delta <1$ is a reasonable sufficient (and likely optimal) condition to obtain solutions of problem (P) in $C^1\left(\overline{\Omega }\right)$.

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