Separation structure of positive radial solutions of a semilinear elliptic equation in Rn

AbstractThis paper deals with the following inter-connected subjects: (i) Separation of positive radial solutions is studied for the elliptic equation Δu+K(|x|)up=0 in Rn. In case r−ℓK(r) is decreasing, with suitable decay rate, to a positive constant as r→∞ for some ℓ>−2, the asymptotic behavior near ∞ of radial solutions is described in detail when n and p are large enough. When the equation has separation structure (any two positive radial solutions do not intersect), the obtained asymptotic behaviors decide natural topology for stability of positive radial solutions regarded as steady states of the corresponding semilinear heat equation. (ii) By using local separation of regular solutions, we establish the existence of a singular solution when every solution with positive initial data at 0 exists globally. (iii) Through Kelvin's transform, separation structure of regular solutions reflects that of singular solutions. We present a new exponent, which is critical in the study of separation and intersection of singular solutions

A priori bounds for positive radial solutions of quasilinear equations of Lane–Emden type

summary:We consider the quasilinear equation $\Delta _p u +K(|x|)u^q=0$, and present the proof of the local existence of positive radial solutions near $0$ under suitable conditions on $K$. Moreover, we provide a priori estimates of positive radial solutions near $\infty$ when $r^{-\ell }K(r)$ for $\ell \ge -p$ is bounded near $\infty$

On a Class of Semilinear Elliptic Equations in Rn

AbstractWe establish that for n⩾3 and p>1, the elliptic equation Δu+K(x)up=0 in Rn possesses separated positive entire solutions of infinite multiplicity, provided that a locally Hölder continuous function K⩾0 in Rn\{0}, satisfies K(x)=O(∣x∣σ) at x=0 for some σ>−2, and K(x)=c∣x∣−2+O(∣x∣−n[log∣x∣]q) near ∞ for some constants c>0 and q>0. In the radial case K(x)=∣x∣l1+∣x∣τ with l>−2 and τ⩾0, or K(x)=∣x∣λ−2(1+∣x∣2)λ/2 with λ>0, we investigate separation phenomena of positive radial solutions, and show that if n and p are large enough, the equation possesses a positive radial solution with initial value α at 0 for each α>0 and a unique positive radial singular solution among which any two solutions do not intersect

Coculture of Marine Streptomyces sp. With Bacillus sp. Produces a New Piperazic Acid-Bearing Cyclic Peptide

Microbial culture conditions in the laboratory, which conventionally involve the cultivation of one strain in one culture vessel, are vastly different from natural microbial environments. Even though perfectly mimicking natural microbial interactions is virtually impossible, the cocultivation of multiple microbial strains is a reasonable strategy to induce the production of secondary metabolites, which enables the discovery of new bioactive natural products. Our coculture of marine Streptomyces and Bacillus strains isolated together from an intertidal mudflat led to discover a new metabolite, dentigerumycin E (1). Dentigerumycin E was determined to be a new cyclic hexapeptide incorporating three piperazic acids, N-OH-Thr, N-OH-Gly, β-OH-Leu, and a pyran-bearing polyketide acyl chain mainly by analysis of its NMR and MS spectroscopic data. The putative PKS-NRPS biosynthetic gene cluster for dentigerumycin E was found in the Streptomyces strain, providing clear evidence that this cyclic peptide is produced by the Streptomyces strain. The absolute configuration of dentigerumycin E was established based on the advanced Marfey's method, ROESY NMR correlations, and analysis of the amino acid sequence of the ketoreductase domain in the biosynthetic gene cluster. In biological evaluation of dentigerumycin E (1) and its chemical derivatives [2-N,16-N-deoxydenteigerumycin E (2) and dentigerumycin methyl ester (3)], only dentigerumycin E exhibited antiproliferative and antimetastatic activities against human cancer cells, indicating that N-OH and carboxylic acid functional groups are essential for the biological activity

Wood Pellet Driven-Biochar Characterization Produced at Different Targeted Pyrolysis Temperatures

The imperative transition to renewable energy drives the need for innovation. Biomass, particularly wood pellets, has demonstrated poor performance in co-firing scenarios. This study employed pyrolysis to convert wood pellets into biochar with improved fuel quality. The biochar production and characterization were investigated at pyrolysis temperatures of 400°C to 500°C. The findings revealed significant improvements: the observed fixed carbon content increased from 67.2% to 78.8%, and the calorific value increased 1.2 times higher within the pyrolysis temperature increased. On the other hand, as the pyrolysis temperature increased from 400°C to 500°C, biochar yields decreased from 49% to 37%. Thermogravimetric (TG) analysis revealed distinct weight loss during heating, illuminating component volatilization and residue accumulation. Furthermore, Fourier transform infrared (FTIR) spectroscopy elucidated structural changes, demonstrating the evolution of cellulose and hemicellulose into aromatic structures. Ultimately, these insights into biochar characteristics informed the optimization of pyrolysis processes, contributing to the production of superior biochar for renewable energy applications. Keywords: biochar, biomass, carbon storage, pyrolysis, wood pelle