## Advances in Differential Equations

- Adv. Differential Equations
- Volume 20, Number 3/4 (2015), 259-298.

### Singular quasilinear elliptic systems and Hölder regularity

Jacques Giacomoni, Ian Schindler, and Peter Takáč

#### 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 $\beta\in (0,1)$.

#### Article information

**Source**

Adv. Differential Equations Volume 20, Number 3/4 (2015), 259-298.

**Dates**

First available in Project Euclid: 4 February 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1423055202

**Mathematical Reviews number (MathSciNet)**

MR3311435

**Zentralblatt MATH identifier**

1312.35081

**Subjects**

Primary: 35J35: Variational methods for higher-order elliptic equations 35J50: Variational methods for elliptic systems 35R05: Partial differential equations with discontinuous coefficients or data

#### Citation

Giacomoni, Jacques; Schindler, Ian; Takáč, Peter. Singular quasilinear elliptic systems and Hölder regularity. Adv. Differential Equations 20 (2015), no. 3/4, 259--298. https://projecteuclid.org/euclid.ade/1423055202