Let \(\Omega\subset \mathbb{R}^N\) (\(N\geq 2\)) be an open set and let \(A:(0,+\infty)\rightarrow (0,+\infty)\), \(B:[0,+\infty)\rightarrow [0,+\infty)\) be \(C^1\)-functions satisfying the following conditions
(i) \(\displaystyle{-1<\inf_{t>0}\frac{tA'(t)}{A(t)}=:m_A\leq M_A:= \sup_{t>0}\frac{tA'(t)}{A(t)}<+\infty}\);
(ii) \(\displaystyle{\sup_{t>0}\frac{B'(t)}{tA(t)}<+\infty}\).
The authors study the local regularity of weak solutions to the degenerate elliptic equation \[ -\operatorname{div}(A(|\nabla u|)\nabla u)+B(|\nabla u|)=f\text{ in } \Omega,\tag{1} \] where \(f\in W^{1,\infty}(\Omega)\), and prove that, for each \(\beta \in [0,1)\) and \(\gamma<\max\{0,N-2\}\) and for each weak solution \(u\in C^{1,\alpha}(\Omega)\cap C^2(\Omega \setminus \{x\in \Omega: \nabla u=0\})\) of equation \((1)\), one has the estimate \[ \int_{B_\rho(x_0)}\frac{A(|\nabla u|)|\nabla u_i|^2}{|x-y|^\gamma|u_i|^\beta}\leq \mathcal{C}, \ \ \ \forall i=1,...,N, \text{ uniformly for } y\in B_\rho(x_0). \] Here, \(\displaystyle{u_i:=\frac{\partial u}{\partial x_i}}\), \(B_\rho(x_0)\) is any ball such that \(B_{2\rho}(x_0)\subset \Omega\), and the constant \(\mathcal{C}\) depends only on \(\gamma,m_A,M_A,\beta,f,\|\nabla u\|_\infty,\rho,x_0\).
As a corollary, the authors derive a local summability property of \((A(|\nabla u|))^{-1}\) under the further condition that \(\inf_{x\in B_{2\rho}(x_0)} |f(u(x))|>0\).
By applying the above regularity results and a weak comparison principle in small domains, the authors also obtain, via moving plane method, monotonicity and symmetry properties of positive solutions to the quasilinear degenerate elliptic problem \[ \left\{\begin{array}{ll} -\operatorname{div}(A(|\nabla u|)\nabla u)+B(|\nabla u|)=f(u), \text{ in }\Omega,\\ u=0 &\text{ on }\Omega \end{array}\right. \] where \(f:(0,+\infty)\rightarrow (0,+\infty)\) is a locally Lipschitz function, the function \(A\) satisfies \((i)\) and some additional monotonicity condition, and the function \(B\) satisfies \((ii)\) and \(B(0)=0\).