scholarly journals Uniqueness of Positive Radial Solutions of the Dirichlet Problems −Δu=up±uqin an Annulus

1997 ◽  
Vol 139 (1) ◽  
pp. 194-217 ◽  
Author(s):  
S.L. Yadava
Keyword(s):  
Radial Solutions ◽  
Dirichlet Problems ◽  
Positive Radial Solutions

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>

Positive radial solutions for Dirichlet problems involving the mean curvature operator in Minkowski space

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Zhiqian He ◽  
Liangying Miao
Keyword(s):  
Mean Curvature ◽  
Fixed Point Index ◽  
Curvature Operator ◽  
Positive Parameter ◽  
Point Index ◽  
Radial Solutions ◽  
Dirichlet Problems ◽  
Positive Radial Solutions ◽  
Mean Curvature Operator ◽  
The Mean

Abstract In this paper, we study the number of classical positive radial solutions for Dirichlet problems of type (P) − d i v ∇ u 1 − | ∇ u | 2 = λ f ( u )   in B 1 , u = 0                     on ∂ B 1 , $$\left\{\begin{aligned}\hfill & -\mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{\nabla u}{\sqrt{1-\vert \nabla u{\vert }^{2}}}\right)=\lambda f(u)\quad \text{in}\enspace {B}_{1},\hfill \\ \hfill & u=0\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \enspace \enspace \enspace \enspace \enspace \enspace \enspace \enspace \enspace \enspace \text{on}\enspace \partial {B}_{1},\enspace \hfill \end{aligned}\right.$$ where λ is a positive parameter, B 1 = { x ∈ R N : | x | < 1 } ${B}_{1}=\left\{x\in {\mathbb{R}}^{N}:\vert x\vert {< }1\right\}$ , f : [0, ∞) → [0, ∞) is a continuous function. Using the fixed point index in a cone, we prove the results on both uniqueness and multiplicity of positive radial solutions of (P).

Uniqueness of positive radial solutions for n-Laplacian Dirichlet problems

2000 ◽  
Vol 130 (6) ◽  
pp. 1405-1416 ◽  
Author(s):  
M. Tang
Keyword(s):  
Dirichlet Problem ◽  
Large Class ◽  
Exponential Growth ◽  
Radial Solutions ◽  
Dirichlet Problems ◽  
Elementary Analysis ◽  
Positive Radial Solutions ◽  
The One

We prove a theorem on the uniqueness of positive radial solutions to a Dirichlet problem of the n-Laplacian in a finite ball of Rn. Our proofs use only elementary analysis based on an identity due to Erbe and Tang. The result can be applied to a large class of nonlinearities, including some polynomials and functions with exponential growth; in particular, the one recently studied by Adimurthi.

scholarly journals The exterior Dirichlet problems of Monge–Ampère equations in dimension two

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Limei Dai
Keyword(s):  
Asymptotic Behavior ◽  
Unit Ball ◽  
Bounded Domain ◽  
Viscosity Solutions ◽  
Sufficient Conditions ◽  
Radial Solutions ◽  
Dirichlet Problems ◽  
Necessary And Sufficient ◽  
Monge Ampere ◽  
Prescribed Asymptotic Behavior

AbstractIn this paper, we study the Monge–Ampère equations $\det D^{2}u=f$ det D 2 u = f in dimension two with f being a perturbation of $f_{0}$ f 0 at infinity. First, we obtain the necessary and sufficient conditions for the existence of radial solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a unit ball. Then, using the Perron method, we get the existence of viscosity solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a bounded domain.

scholarly journals Radial solutions of Dirichlet problems with concave–convex nonlinearities

2011 ◽  
Vol 74 (7) ◽  
pp. 2720-2738 ◽  
Author(s):  
Francesca Dalbono ◽  
Walter Dambrosio
Keyword(s):  
Radial Solutions ◽  
Dirichlet Problems
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