Singular quasilinear elliptic systems and Hölder regularity

Abstract

We investigate the following singular quasilinear elliptic system, \begin{equation} \nonumber \tag{P} \left. \begin{alignedat}{2} - \Delta_p u & = \frac{1}{ u^{a_1} v^{b_1} }\, \quad\mbox{ in }\,\Omega ;\quad u\vert_{\partial\Omega} & = 0 ,\quad u > 0\quad\mbox{ in }\,\Omega , \\ - \Delta_q v & = \frac{1}{ v^{a_2} u^{b_2} }\, \quad\mbox{ in }\,\Omega ;\quad v\vert_{\partial\Omega} & = 0 ,\quad v > 0\quad\mbox{ in }\,\Omega , \end{alignedat} \quad \right\} \end{equation} where $\Omega$ is an open bounded domain with smooth boundary, $1 < p, q < \infty$, and the numbers $a_1, a_2, b_1, b_2 > 0$ satisfy certain upper bounds. We employ monotonicity methods in order to prove the existence and uniqueness of a pair of positive solutions to (P). While following a standard fixed point approach with ordered pairs of sub- and super solutions, we need to prove a new regularity result of independent interest for solution pairs to problem (P) in $C^{0,\beta}(\overline{\Omega})$ with some $\beta\in (0,1)$.