scholarly journals On stable solutions of the weighted Lane-Emden equation involving Grushin operator

2021 ◽  
Vol 6 (3) ◽  
pp. 2623-2635
Author(s):  
Yunfeng Wei ◽  
◽  
Hongwei Yang ◽  
Hongwang Yu ◽  
Keyword(s):  
Emden Equation ◽  
Grushin Operator ◽  
Stable Solutions

scholarly journals On stable solutions of the fractional Hénon–Lane–Emden equation

2016 ◽  
Vol 18 (05) ◽  
pp. 1650005 ◽  
Author(s):  
Mostafa Fazly ◽  
Juncheng Wei
Keyword(s):  
Emden Equation ◽  
Stable Solutions

We derive monotonicity formulae for solutions of the fractional Hénon–Lane–Emden equation [Formula: see text] when [Formula: see text], [Formula: see text] and [Formula: see text]. Then, we apply these formulae to classify stable solutions of the above equation.

STABLE SOLUTIONS TO THE STATIC CHOQUARD EQUATION

2020 ◽  
Vol 102 (3) ◽  
pp. 471-478
Author(s):  
PHUONG LE
Keyword(s):  
Weak Solution ◽  
Liouville Theorem ◽  
Choquard Equation ◽  
High Dimensions ◽  
Emden Equation ◽  
Stable Solutions ◽  
Image Position

This paper is concerned with the static Choquard equation $$\begin{eqnarray}-\unicode[STIX]{x1D6E5}u=\bigg(\frac{1}{|x|^{N-\unicode[STIX]{x1D6FC}}}\ast |u|^{p}\bigg)|u|^{p-2}u\quad \text{in }\mathbb{R}^{N},\end{eqnarray}$$ where $N,p>2$ and $\max \{0,N-4\}<\unicode[STIX]{x1D6FC}<N$. We prove that if $u\in C^{1}(\mathbb{R}^{N})$ is a stable weak solution of the equation, then $u\equiv 0$. This phenomenon is quite different from that of the local Lane–Emden equation, where such a result only holds for low exponents in high dimensions. Our result is the first Liouville theorem for Choquard-type equations with supercritical exponents and $\unicode[STIX]{x1D6FC}\neq 2$.

scholarly journals Finite Morse index solutions of the Hénon Lane–Emden equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Abdellaziz Harrabi ◽  
Cherif Zaidi
Keyword(s):  
Morse Index ◽  
Monotonicity Formula ◽  
Liouville Type ◽  
Emden Equation ◽  
Type Identity ◽  
Liouville Type Theorems ◽  
Whole Space ◽  
Stable Solutions ◽  
Pohozaev Type Identity

Abstract In this paper, we are concerned with Liouville-type theorems of the Hénon Lane–Emden triharmonic equations in whole space. We prove Liouville-type theorems for solutions belonging to one of the following classes: stable solutions and finite Morse index solutions (whether positive or sign-changing). Our proof is based on a combination of the Pohozaev-type identity, monotonicity formula of solutions and a blowing down sequence.

scholarly journals Liouville type theorems for stable solutions of elliptic system involving the Grushin operator

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Foued Mtiri
Keyword(s):  
Elliptic System ◽  
Stable Solution ◽  
Integral Estimate ◽  
Liouville Type ◽  
Grushin Operator ◽  
Degenerate Elliptic System ◽  
Liouville Type Theorems ◽  
Degenerate Elliptic ◽  
Stable Solutions ◽  
Alpha Beta

<p style='text-indent:20px;'>We examine the following degenerate elliptic system:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\Delta_{s} u \! = \! v^p, \quad -\Delta_{s} v\! = \! u^\theta, \;\; u, v&gt;0 \;\;\mbox{in }\; \mathbb{R}^N = \mathbb{R}^{N_1}\times \mathbb{R}^{N_2}, \quad\mbox{where}\;\; s \geq 0\;\; \mbox{and} \;\;p, \theta \!&gt;\!0. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>We prove that the system has no stable solution provided <inline-formula><tex-math id="M1">\begin{document}$ p, \theta &gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ N_s: = N_1+(1+s)N_2&lt; 2 + \alpha + \beta, $\end{document}</tex-math></inline-formula> where</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \alpha = \frac{2(p+1)}{p\theta - 1} \quad\mbox{and} \quad \beta = \frac{2(\theta +1)}{p\theta - 1}. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>This result is an extension of some results in [<xref ref-type="bibr" rid="b15">15</xref>]. In particular, we establish a new integral estimate for <inline-formula><tex-math id="M3">\begin{document}$ u $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ v $\end{document}</tex-math></inline-formula> (see Proposition 1.1), which is crucial to deal with the case <inline-formula><tex-math id="M5">\begin{document}$ 0 &lt; p &lt; 1. $\end{document}</tex-math></inline-formula></p>

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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