scholarly journals Liouville theorems for stable solutions of the weighted Lane-Emden system

2017 ◽  
Vol 37 (1) ◽  
pp. 265-279 ◽  
Author(s):  
Hatem Hajlaoui ◽  
◽  
Abdellaziz Harrabi ◽  
Foued Mtiri ◽  
◽  
...  
Keyword(s):  
Liouville Theorems ◽  
Stable Solutions

The Influence of Some Factors on Spirolactone Stability in Solution

2008 ◽  
Vol 59 (7) ◽  
Author(s):  
Daniela Lucia Muntean ◽  
Silvia Imre ◽  
Cosmina Voda
Keyword(s):  
High Performance Liquid Chromatography ◽  
Liquid Chromatography ◽  
Uv Irradiation ◽  
High Performance ◽  
Degradation Products ◽  
Simultaneous Action ◽  
Ethanolic Solution ◽  
Degradation Processes ◽  
Stable Solutions ◽  
Preformulation Study

The influence of some factors on spironolactone stability in solution was studied, by applying high-performance liquid chromatography, as a part of a pharmaceutical preformulation study in order to obtain a spironolactone solution for alopecia treatment. Solutions of 1 mg/ml spironolactone in aqueous ethanolic solution 1 : 1 and in 20 mM cyclodextrines solutions (b-, hydroxi-b- and methyl-b-cyclodextrine) was used, maintained at 8 and 22 �C, protected from light and after UV irradiation at 254 nm. The main degradation products were 7a-thiospirolactone and canrenone. The most stable solutions were the alcoholic ones and with methyl-beta-cyclodextrine, but the simultaneous action of temperature and UV irradiation allowed degradation processes after one hour of exposure, more aggressive in the presence of methyl-beta-cyclodextrine. In conclusion, for alopecia treatment with spironolactone a 1 mg/mL ethanolic solution could be used and it is recommendable the protection of treated zone.

scholarly journals Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Xavier Cabré ◽  
Pietro Miraglio ◽  
Manel Sanchón
Keyword(s):  
Dirichlet Problem ◽  
Optimal Condition ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Nonlinear Equations ◽  
Open Problem ◽  
Optimal Range ◽  
Optimal Regularity ◽  
Smooth Bounded Domain ◽  
Stable Solutions

AbstractWe consider the equation {-\Delta_{p}u=f(u)} in a smooth bounded domain of {\mathbb{R}^{n}}, where {\Delta_{p}} is the p-Laplace operator. Explicit examples of unbounded stable energy solutions are known if {n\geq p+\frac{4p}{p-1}}. Instead, when {n<p+\frac{4p}{p-1}}, stable solutions have been proved to be bounded only in the radial case or under strong assumptions on f. In this article we solve a long-standing open problem: we prove an interior {C^{\alpha}} bound for stable solutions which holds for every nonnegative {f\in C^{1}} whenever {p\geq 2} and the optimal condition {n<p+\frac{4p}{p-1}} holds. When {p\in(1,2)}, we obtain the same result under the nonsharp assumption {n<5p}. These interior estimates lead to the boundedness of stable and extremal solutions to the associated Dirichlet problem when the domain is strictly convex. Our work extends to the p-Laplacian some of the recent results of Figalli, Ros-Oton, Serra, and the first author for the classical Laplacian, which have established the regularity of stable solutions when {p=2} in the optimal range {n<10}.

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