Equations involving fractional Laplacian operator: Compactness and application

In this paper, we consider the following problem involving fractional Laplacian operator: \begin{equation}\label{eq:0.1} (-\Delta)^{\alpha} u= |u|^{2^*_\alpha-2-\varepsilon}u + \lambda u\,\, {\rm in}\,\, \Omega,\quad u=0 \,\, {\rm on}\, \, \partial\Omega, \end{equation} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $\varepsilon\in [0, 2^*_\alpha-2)$, $0<\alpha<1,\, 2^*_\alpha = \frac {2N}{N-2\alpha}$. We show that for any sequence of solutions $u_n$ of \eqref{eq:0.1} corresponding to $\varepsilon_n\in [0, 2^*_\alpha-2)$, satisfying $\|u_n\|_{H}\le C$ in the Sobolev space $H$ defined in \eqref{eq:1.1a}, $u_n$ converges strongly in $H$ provided that $N>6\alpha$ and $\lambda>0$. An application of this compactness result is that problem \eqref{eq:0.1} possesses infinitely many solutions under the same assumptions.Comment: 34 page

Biohydrogen Production by the Hyperthermophilic Bacterium Thermotoga neapolitana

ABSTRACT Thermotoga neapolitana can use different sources of carbon and nitrogen for growth and produces biological hydrogen. Sources of carbon (glucose, sucrose, xylose, xylan, cellulose, cellobiose, starch, corn starch, beet bulp pellet, and rice flour) and nitrogen (yeast extract, fish meal, cottonseed meal, canola meal, linseed meal, and soybean meal) were compared. In the carbon studies, glucose, sucrose, rice flour, and xylan produced similar levels of hydrogen. In the nitrogen studies, Trypticase combined with alternative nitrogen sources can efficiently increases the yield of hydrogen produced by Thermotoga neapolitana. Yeast extract with trypticase as the dual nitrogen sources produced significantly increased concentration of hydrogen than the other combinations tested. Soybean meal and canola meal were second choices as alternative nitrogen sources. Sucrose and rice flour were promising carbon sources to replace glucose, and soybean meals was a promising nitrogen source to replace yeast extract for Thermotoga neapolitana. Thermotoga neapolitana can utilize rice flour as sole carbon source, and soybean meal as one of nitrogen sources to produce hydrogen. Uniform design was used as experimental design to optimize the fermentation medium. The optimized medium was composed of 9 g/L rice flour, 4.5 g/L soybean meal, and 4.5 g/L trypticase. The hydrogen concentration for this optimized medium was 0.07083±0.006198 g H2/L medium or 35.42±3.10 mmol H2/L medium. The increased hydrogen concentration from control medium to optimized medium was 21.6%. Thermotoga neapolitana ferments glucose as carbon source to produce acetate, carbon dioxide, and hydrogen as major products. The exponential phase of the bacterial growth was between 2hrs and 10hrs of incubation time. The maximum cell mass concentration was reached after 10hrs of incubation. The stationary phase lasted for 10hrs, and the death phase began at 20hrs. The pH of broth decreased during the bacterium growth, which may inhibit hydrogen production. The maximum hydrogen partial pressure in this study was 45 kPa at 77°C, and hydrogen partial pressure might inhibit the hydrogen production in this batch fermentation, an estimate of the critical hydrogen partial pressure of 38kPa was calculated for the batch fermentation of Thermotoga neapolitana at 77°C. The maximum specific growth rate ( ) of Thermotoga neapolitana with glucose as carbon source was 0.94hr-1 at 77°C. The Monod half saturation constant (KS) was 0.57 g/L, the observed biomass yield from substrate was 0.25 g/g glucose or 44.59 g/mol glucose, the observed hydrogen yield from substrate was 0.028 g/glucose or 2.50 mol H2 /mol glucose, and the observed hydrogen yield from biomass was 0.114 g/g dry weight. When glucose concentration was 5.0 g/L, the doubling time was 0.84hr or 49mins

Evaluating Agricultural Banking Efficiency Using the Fourier Flexible Functional Form

This study applied more flexible cost functional form, Fourier Flexible Functional Form, and tested the validity of the Translog cost functional form as to estimate the cost function incorporating risk and loan's quality for banking industry. Meanwhile, the study extended four different cost efficiency measures for banking industry not only among different sized banks but also between commercial banks and agricultural banks. And thereafter, by evaluating these efficiency measures, banks will identify sources of inefficiency, which should aid banks in developing approaches to improve their operational policies, procedures, and performance.Agricultural Finance,