scholarly journals Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations

2021 ◽  
Vol 41 (9) ◽  
pp. 4185
Author(s):  
Mostafa Fazly ◽  
Yuan Li
Keyword(s):  
Elliptic Equations ◽  
Partial Regularity ◽  
Liouville Theorems ◽  
Stable Solutions ◽  
Anisotropic Elliptic Equations

scholarly journals Extremal solution and Liouville theorem for anisotropic elliptic equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yuan Li
Keyword(s):  
Convex Function ◽  
Elliptic Equations ◽  
Support Function ◽  
Morse Index ◽  
Dirichlet Boundary ◽  
Liouville Theorem ◽  
Extremal Solution ◽  
Stable Solutions ◽  
Dirichlet Boundary Problem ◽  
Anisotropic Elliptic Equations

<p style='text-indent:20px;'>We study the quasilinear Dirichlet boundary problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \nonumber \begin{cases} -Qu = \lambda e^{u}, \text{in}~~ \Omega, \\ u = 0, \qquad \;~~\text{on}~~~~ \partial\Omega, \end{cases} \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula> is a parameter, <inline-formula><tex-math id="M2">\begin{document}$ \Omega\subset\mathbb{R}^{N} $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M3">\begin{document}$ N\geq2 $\end{document}</tex-math></inline-formula>) is a bounded domain, and the operator <inline-formula><tex-math id="M4">\begin{document}$ Q $\end{document}</tex-math></inline-formula>, known as Finsler-Laplacian or anisotropic Laplacian, is defined by</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ Qu: = \sum\limits_{i = 1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{\xi_{i}}(\nabla u)). $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Here, <inline-formula><tex-math id="M5">\begin{document}$ F_{\xi_{i}} = \frac{\partial F}{\partial\xi_{i}}(\xi) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ F: \mathbb{R}^{N}\rightarrow [0, +\infty) $\end{document}</tex-math></inline-formula> is a convex function of <inline-formula><tex-math id="M7">\begin{document}$ C^{2}(\mathbb{R}^{N}\setminus\{0\}) $\end{document}</tex-math></inline-formula>, and satisfies certain assumptions. We derive the existence of extremal solution and obtain that it is regular, if <inline-formula><tex-math id="M8">\begin{document}$ N\leq9 $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>We also concern the Hénon type anisotropic Liouville equation, </p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ -Qu = (F^{0}(x))^{\alpha}e^{u} ~~\text{in} ~~\mathbb{R}^{N}, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M9">\begin{document}$ \alpha&gt;-2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ N\geq2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ F^{0} $\end{document}</tex-math></inline-formula> is the support function of <inline-formula><tex-math id="M12">\begin{document}$ K: = \{x\in\mathbb{R}^{N}:F(x)&lt;1\} $\end{document}</tex-math></inline-formula>. We obtain the Liouville theorem for stable solutions and finite Morse index solutions for <inline-formula><tex-math id="M13">\begin{document}$ 2\leq N&lt;10+4\alpha $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M14">\begin{document}$ 3\leq N&lt;10+4\alpha^{-} $\end{document}</tex-math></inline-formula> respectively, where <inline-formula><tex-math id="M15">\begin{document}$ \alpha^{-} = \min\{\alpha, 0\} $\end{document}</tex-math></inline-formula>.</p>

scholarly journals Partial regularity of stable solutions to the fractional Geľfand-Liouville equation

2021 ◽  
Vol 10 (1) ◽  
pp. 1316-1327
Author(s):  
Ali Hyder ◽  
Wen Yang
Keyword(s):  
Liouville Equation ◽  
Weak Solutions ◽  
Partial Regularity ◽  
Singular Set ◽  
Stable Solutions ◽  
Gelfand Problem

Abstract We analyze stable weak solutions to the fractional Geľfand problem ( − Δ ) s u = e u i n Ω ⊂ R n . $$\begin{array}{} \displaystyle (-{\it\Delta})^su = e^u\quad\mathrm{in}\quad {\it\Omega}\subset\mathbb{R}^n. \end{array}$$ We prove that the dimension of the singular set is at most n − 10s.

scholarly journals Entropy and renormalized solutions for some nonlinear anisotropic elliptic equations with variable exponents and L 1-data

2021 ◽  
Vol 7 (2) ◽  
pp. 277-298
Author(s):  
Mostafa El Moumni ◽  
Deval Sidi Mohamed
Keyword(s):  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Variable Exponents ◽  
Renormalized Solutions ◽  
Nonlinear Elliptic ◽  
Anisotropic Elliptic Equations

Abstract We prove in this paper some existence and unicity results of entropy and renormalized solutions for some nonlinear elliptic equations with general anisotropic diffusivities and variable exponents. The data are assumed to be merely integrable.

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