Existence and regularity of solutions to a quasilinear elliptic problem involving variable sources

The authors of this paper prove the existence and regularity results for the homogeneous Dirichlet boundary value problem to the equation \(-\operatorname{div}(|\nabla u|^{p-2}\nabla u)=\frac{f(x)}{u ^{\alpha(x)}}\) with \(f\in L^{m}(\Omega)\) (\(m\geqslant1\)) and \(\alpha(x)>0\). Due to the nonlinearity of a p-Laplace operator and the anisotropic variable exponent \(\alpha(x)\), some classical methods may not directly be applied to our problem. In this paper, we construct a suitable test function and apply the Leray-Schauder fixed point theorem to prove the existence of positive solutions with necessary a priori estimate and compact argument. Furthermore, we also discuss the relationship among the regularity of solutions, the summability of f and the value of \(\alpha(x)\).