In this paper, we consider the indefinite fractional elliptic problem. A corresponding Liouville-type theorem for the indefinite fractional elliptic equations is established. Furthermore, we obtain a priori bound for solutions in a bounded domain by blowing-up and re-scaling. We also classify the solutions of some degenerate elliptic equation originated from fractional Laplacian.

In this article, we study some quantitative unique continuation properties of solutions to second order elliptic equations with singular lower order terms. First, we quantify the strong unique continuation property by estimating the maximal vanishing order of solutions. That is, when u is a non-trivial solution to ∆u + W · ∇u + V u = 0 in some open, connected subset of R n , where n ≥ 3, we characterize the vanishing order of solutions in terms of the norms of V and W in their respective Lebesgue spaces. Then, using these maximal order of vanishing estimates, we establish quantitative unique continuation at infinity results for solutions to ∆u + W · ∇u + V u = 0 in R n . The main tools in our work are new versions of L p → L q Carleman estimates for a range of p-and q-values.

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Materials capable of extracting gold from complex sources, especially electronic waste (e-waste), are needed for gold resource sustainability and effective e-waste recycling. However, it remains challenging to achieve high extraction capacity and precise selectivity if only a trace amount of gold is present along with other metallic elements . Here we report an approach based on reduced graphene oxide (rGO) which provides an ultrahigh capacity and selective extraction of gold ions present in ppm concentrations (>1000 mg of gold per gram of rGO at 1 ppm). The excellent gold extraction performance is accounted to the graphene areas and oxidized regions of rGO. The graphene areas spontaneously reduce gold ions to metallic gold, and the oxidized regions allow good dispersibility of the rGO material so that efficient adsorption and reduction of gold ions at the graphene areas can be realized. By controlling the protonation of the oxidized regions of rGO, gold can be extracted exclusively, without contamination by the other 14 co-existing elements typically present in e-waste. These findings are further exploited to demonstrate recycling gold from real-world e-waste with good scalability and economic viability, as exemplified by using rGO membranes in a continuous flow-through process.