Uniqueness theorems through the method of moving spheres

DOI10.1215/S0012-7094-95-08016-8zbMath0846.35050MaRDI QIDQ1913363

Meijun Zhu, Yan-yan Li

Publication date: 6 October 1996

Published in: Duke Mathematical Journal (Search for Journal in Brave)

zbMATH Keywords

nonlinear boundary conditioncritical Sobolev exponent

Mathematics Subject Classification ID

Nonlinear boundary value problems for linear elliptic equations (35J65)

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Cites Work

• Unnamed Item
• Unnamed Item
• Problèmes de Neumann non linéaires sur les variétés riemanniennes
• Symmetry and related properties via the maximum principle
• Classification of solutions of some nonlinear elliptic equations
• Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary
• Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition
• Asymptotic radial symmetry for solutions of \(\Delta u+e^ u=0\) in a punctured disc
• Conformal metrics with prescribed mean curvature on the boundary
• Prescribing scalar curvature on \(\mathbb{S}^ n\) and related problems. I
• Symmetry properties of positive solutions to some elliptic equations with nonlinear boundary conditions
• On the solutions to some elliptic equations with nonlinear Neumann boundary conditions
• Positive harmonic functions on the upper half space satisfying a nonlinear boundary condition
• A symmetry problem in potential theory
• The conjectures on conformal transformations of Riemannian manifolds
• Uniqueness theorems on conformal deformation of metrics, sobolev inequalities, and an eigenvalue estimate
• Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth
• Global and local behavior of positive solutions of nonlinear elliptic equations
• Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains
• Uniform estimates and blow–up behavior for solutions of −δ(u)=v(x)e^uin two dimensions
• The principal eigenvalue and maximum principle for second‐order elliptic operators in general domains
• On –δu = K(x)u⁵ in ℝ³
• Symmetry of the blow‐up set of a porous medium type equation
• A necessary and sufficient condition for the nirenberg problem
• On the method of moving planes and the sliding method