Existence and uniqueness of nontrivial radial solutions for k-Hessian equations

2020 ◽  
Vol 492 (1) ◽  
pp. 124439
Author(s):  
Xuemei Zhang
Keyword(s):  
Existence And Uniqueness ◽  
Radial Solutions ◽  
Hessian Equations

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>

Sharp estimates of semistable radial solutions of k-Hessian equations

2019 ◽  
Vol 150 (4) ◽  
pp. 2083-2115 ◽  
Author(s):  
Miguel Angel Navarro ◽  
Justino Sánchez
Keyword(s):  
Sobolev Space ◽  
Weighted Sobolev Space ◽  
Extremal Solution ◽  
Nonincreasing Function ◽  
Pointwise Estimates ◽  
Radial Solutions ◽  
Sharp Estimates ◽  
Dirichlet Data ◽  
Hessian Equations ◽  
Hessian Operator

AbstractWe consider semistable, radially symmetric and increasing solutions of Sk(D2u) = g(u) in the unit ball of ℝn, where Sk(D2u) is the k-Hessian operator of u and g ∈ C1 is a general positive nonlinearity. We establish sharp pointwise estimates for such solutions in a proper weighted Sobolev space, which are optimal and do not depend on the specific nonlinearity g. As an application of these results, we obtain pointwise estimates for the extremal solution and its derivatives (up to order three) of the equation Sk(D2u) = λg(u), posed in B1, with Dirichlet data $u\arrowvert _{B_1}=0$, where g is a continuous, positive, nonincreasing function such that lim t→−∞g(t)/|t|k = +∞.

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