• Bulletin of the Korean Mathematical Society
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• v.48 no.5
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• pp.1111-1117
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• 2011

Using the moving plane method, we establish the radial symmetry of topological one-vortex solutions of a semilinear elliptic system arising from the self-dual Maxwell-Chern-Simons O(3) model.

• Communications of the Korean Mathematical Society
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• v.19 no.2
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• pp.283-291
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• 2004

In this paper we show the radial symmetry of topological one-vortex solutions in the Maxwell-Chern-Simons-Higgs Model.

• Communications of the Korean Mathematical Society
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• v.10 no.1
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• pp.133-135
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• 1995

In this note, we will investigate the radial symmetry of some kind of solutions of nonlinear ellipitic equations $$ \Delta U = f(U) $$ $$ (1.1) U = 0 in B $$ $$ U \in C^2 (\bar{B}) on \partial B$$ Here f is $C^1$ and B denotes a n-dimensional unit ball in $R^n$.

• Journal of the Korean Mathematical Society
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• v.37 no.5
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• pp.751-761
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• 2000

Symmetry properties of positive solutions for semilinear elliptic problems in n are considered. We give a symmetry result for the problem in the feneral case, and then derive various results for certain classes of demilinear elliptic equations. We employ the moving plane method based on the maximum principle on unbounded domains to obtain the result on symmetry.

• Journal of the Korean Mathematical Society
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• v.53 no.2
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• pp.305-313
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• 2016

In this paper, we give several sufficient conditions ensuring that any positive radial solution (u, v) of the following ${\gamma}$-Laplacian systems in the whole space ${\mathbb{R}}^n$ has the components symmetry property $u{\equiv}v$ $$\{\array{-div({\mid}{\nabla}u{\mid}^{{\gamma}-2}{\nabla}u)=f(u,v)\text{ in }{\mathbb{R}}^n,\\-div({\mid}{\nabla}v{\mid}^{{\gamma}-2}{\nabla}v)=g(u,v)\text{ in }{\mathbb{R}}^n.}$$ Here n > ${\gamma}$, ${\gamma}$ > 1. Thus, the systems will be reduced to a single ${\gamma}$-Laplacian equation: $$-div({\mid}{\nabla}u{\mid}^{{\gamma}-2}{\nabla}u)=f(u)\text{ in }{\mathbb{R}}^n$$. Our proofs are based on suitable comparation principle arguments, combined with properties of radial solutions.

• Journal of the Korean Mathematical Society
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• v.61 no.3
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• pp.445-459
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• 2024

In this paper, by establishing the direct method of moving planes for the fractional Laplacian system with positive-negative mixed powers, we obtain the radial symmetry and monotonicity of the positive solutions for the fractional Laplacian systems with positive-negative mixed powers in the whole space. We also give two special cases.

• Journal of the Korean Mathematical Society
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• v.58 no.6
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• pp.1449-1460
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• 2021

In this paper, we consider the following nonlocal fractional Laplacian equation with a singular nonlinearity (-∆)^su(x) = λu^β (x) + a₀u^-γ (x), x ∈ ℝⁿ, where 0 < s < 1, γ > 0, $1<{\beta}{\leq}\frac{n+2s}{n-2s}$, λ > 0 are constants and a₀ ≥ 0. We use a direct method of moving planes which introduced by Chen-Li-Li to prove that positive solutions u(x) must be radially symmetric and monotone increasing about some point in ℝⁿ.