Classification of the structure of positive radial solutions to $$\Delta u + K(|x|)u^p = 0$$ inR n

1993 ◽  
Vol 124 (3) ◽  
pp. 239-259 ◽  
Author(s):  
Eiji Yanagida ◽  
Shoji Yotsutani
Keyword(s):  
Radial Solutions ◽  
Positive Radial Solutions

Coercive elliptic systems with gradient terms

2017 ◽  
Vol 6 (2) ◽  
pp. 165-182 ◽  
Author(s):  
Roberta Filippucci ◽  
Federico Vinti
Keyword(s):  
Blow Up ◽  
Elliptic Systems ◽  
Continuous Functions ◽  
Necessary Condition ◽  
Bounded Solutions ◽  
Radial Solutions ◽  
Existence Of A Solution ◽  
Content Type ◽  
Positive Radial Solutions

AbstractIn this paper we give a classification of positive radial solutions of the following system:$\Delta u=v^{m},\quad\Delta v=h(|x|)g(u)f(|\nabla u|),$in the open ball ${B_{R}}$, with ${m>0}$, and f, g, h nonnegative nondecreasing continuous functions. In particular, we deal with both explosive and bounded solutions. Our results involve, as in [27], a generalization of the well-known Keller–Osserman condition, namely, ${\int_{1}^{\infty}(\int_{0}^{s}F(t)\,dt)^{-m/(2m+1)}\,ds<\infty}$, where ${F(t)=\int_{0}^{t}f(s)\,ds}$. Moreover, in the second part of the paper, the p-Laplacian version, given by ${\Delta_{p}u=v^{m}}$, ${\Delta_{p}v=f(|\nabla u|)}$, is treated. When ${p\geq 2}$, we prove a necessary condition for the existence of a solution with at least a blow up component at the boundary, precisely ${\int_{1}^{\infty}(\int_{0}^{s}F(t)\,dt)^{-m/(mp+p-1)}s^{(p-2)(p-1)/(mp+p-1)}% \,ds<\infty}$.

scholarly journals Classification of positive radial solutions to a weighted biharmonic equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yuhao Yan
Keyword(s):  
Biharmonic Equation ◽  
Type Inequality ◽  
Removable Singularity ◽  
Radial Solution ◽  
Order Equation ◽  
Radial Solutions ◽  
Best Constant ◽  
Positive Radial Solutions ◽  
Alpha Beta

<p style='text-indent:20px;'>In this paper, we consider the weighted fourth order equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \Delta(|x|^{-\alpha}\Delta u)+\lambda \text{div}(|x|^{-\alpha-2}\nabla u)+\mu|x|^{-\alpha-4}u = |x|^\beta u^p\quad \text{in} \quad \mathbb{R}^n \backslash \{0\}, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ n\geq 5 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ -n&lt;\alpha&lt;n-4 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ p&gt;1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ (p,\alpha,\beta,n) $\end{document}</tex-math></inline-formula> belongs to the critical hyperbola</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \frac{n+\alpha}{2}+\frac{n+\beta}{p+1} = n-2. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>We prove the existence of radial solutions to the equation for some <inline-formula><tex-math id="M5">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>. On the other hand, let <inline-formula><tex-math id="M7">\begin{document}$ v(t): = |x|^{\frac{n-4-\alpha}{2}}u(|x|) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ t = -\ln |x| $\end{document}</tex-math></inline-formula>, then for the radial solution <inline-formula><tex-math id="M9">\begin{document}$ u $\end{document}</tex-math></inline-formula> with non-removable singularity at origin, <inline-formula><tex-math id="M10">\begin{document}$ v(t) $\end{document}</tex-math></inline-formula> is a periodic function if <inline-formula><tex-math id="M11">\begin{document}$ \alpha \in (-2,n-4) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M12">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M13">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> satisfy some conditions; while for <inline-formula><tex-math id="M14">\begin{document}$ \alpha \in (-n,-2] $\end{document}</tex-math></inline-formula>, there exists a radial solution with non-removable singularity and the corresponding function <inline-formula><tex-math id="M15">\begin{document}$ v(t) $\end{document}</tex-math></inline-formula> is not periodic. We also get some results about the best constant and symmetry breaking, which is closely related to the Caffarelli-Kohn-Nirenberg type inequality.</p>

scholarly journals Weighted fourth order elliptic problems in the unit ball

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zongming Guo ◽  
Fangshu Wan
Keyword(s):  
Asymptotic Behavior ◽  
Singular Point ◽  
Unit Ball ◽  
Fourth Order ◽  
Existence And Uniqueness ◽  
Weighted Sobolev Spaces ◽  
Radial Solutions ◽  
Dirichlet Problems ◽  
Uniqueness Results ◽  
Positive Radial Solutions

<p style='text-indent:20px;'>Existence and uniqueness of positive radial solutions of some weighted fourth order elliptic Navier and Dirichlet problems in the unit ball <inline-formula><tex-math id="M1">\begin{document}$ B $\end{document}</tex-math></inline-formula> are studied. The weights can be singular at <inline-formula><tex-math id="M2">\begin{document}$ x = 0 \in B $\end{document}</tex-math></inline-formula>. Existence of positive radial solutions of the problems is obtained via variational methods in the weighted Sobolev spaces. To obtain the uniqueness results, we need to know exactly the asymptotic behavior of the solutions at the singular point <inline-formula><tex-math id="M3">\begin{document}$ x = 0 $\end{document}</tex-math></inline-formula>.</p>

Sign in / Sign up

Export Citation Format

Share Document