Abstract

Let \(v\) and \(w\) be two weights, i.e., nonnegative locally integrable functions, defined on a bounded domain \(\Omega \subset{\mathbf R}^n , n\geq 3\). The authors study explicit sufficient conditions on \(v\) and \(w\) for the validity of the following two-weight Sobolev’s inequality \[ \left(\int_\Omega|u(x)|^q\omega (x) dx\right)^{{1}/{q}}\leq C \left(\int_\Omega|\triangledown u(x)|^p v(x) dx\right)^{{1}/{p}}, \] where \(1 < p \leq q <\infty\) and \(u\) varies in \(C^\infty_c (\Omega)\), the class of compactly supported and infinitely differentiable functions on \(\Omega\). The results not only improve and unify several kinds of inequalities, but also have been applied to study the Green function, existence, uniqueness and regularity of generalized solutions for the following second-order singular elliptic operator: \[ Lu = - \sum^n_{i,j=1}D_i (a^{ij} D_j u) + \sum^n_{j=1}b^j D_j u + cu, \] where the coefficients \(a^{ij} , 1\leq i, j \leq n, b^j , 1\leq j\leq n\), and \(c\) are real measurable functions on a bounded domain \(\Omega \subset{\mathbf R}^n\) that satisfies an exterior sphere condition uniformly.