2008, 1(2): 225-233. doi: 10.3934/dcdss.2008.1.225

A priori estimate for the Nirenberg problem

1. 

Department of Mathematics, Yeshiva University, 500 W 185th Street, New York, NY 10033

2. 

Department of Applied Mathematics, University of Colorado at Boulder, Boulder, CO 80309-0524

Received  October 2006 Revised  September 2007 Published  March 2008

We establish a priori estimate for solutions to the prescribing Gaussian curvature equation

$ - \Delta u + 1 = K(x) e^{2u}, x \in S^2,$    (1)

for functions $K(x)$ which are allowed to change signs. In [16], Chang, Gursky and Yang obtained a priori estimate for the solution of (1) under the condition that the function K(x) be positive and bounded away from 0. This technical assumption was used to guarantee a uniform bound on the energy of the solutions. The main objective of our paper is to remove this well-known assumption. Using the method of moving planes in a local way, we are able to control the growth of the solutions in the region where K is negative and in the region where K is small and thus obtain a priori estimate on the solutions of (1) for general functions K with changing signs.

Citation: Wenxiong Chen, Congming Li. A priori estimate for the Nirenberg problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 225-233. doi: 10.3934/dcdss.2008.1.225
[1]

Li Ma, Lin Zhao. Regularity for positive weak solutions to semi-linear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (3) : 631-643. doi: 10.3934/cpaa.2008.7.631

[2]

Paul Sacks, Mahamadi Warma. Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 761-787. doi: 10.3934/dcds.2014.34.761

[3]

Meixia Dou. A direct method of moving planes for fractional Laplacian equations in the unit ball. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1797-1807. doi: 10.3934/cpaa.2016015

[4]

Jesus Idelfonso Díaz, Jean Michel Rakotoson. On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1037-1058. doi: 10.3934/dcds.2010.27.1037

[5]

Houda Mokrani. Semi-linear sub-elliptic equations on the Heisenberg group with a singular potential. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1619-1636. doi: 10.3934/cpaa.2009.8.1619

[6]

Út V. Lê. Contraction-Galerkin method for a semi-linear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (1) : 141-160. doi: 10.3934/cpaa.2010.9.141

[7]

Nguyen Thieu Huy, Vu Thi Ngoc Ha, Pham Truong Xuan. Boundedness and stability of solutions to semi-linear equations and applications to fluid dynamics. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2103-2116. doi: 10.3934/cpaa.2016029

[8]

Guangyue Huang, Wenyi Chen. Uniqueness for the solution of semi-linear elliptic Neumann problems in $\mathbb R^3$. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1269-1273. doi: 10.3934/cpaa.2008.7.1269

[9]

Audric Drogoul, Gilles Aubert. The topological gradient method for semi-linear problems and application to edge detection and noise removal. Inverse Problems & Imaging, 2016, 10 (1) : 51-86. doi: 10.3934/ipi.2016.10.51

[10]

Y. Kabeya, Eiji Yanagida, Shoji Yotsutani. Canonical forms and structure theorems for radial solutions to semi-linear elliptic problems. Communications on Pure & Applied Analysis, 2002, 1 (1) : 85-102. doi: 10.3934/cpaa.2002.1.85

[11]

Pengyan Wang, Pengcheng Niu. A direct method of moving planes for a fully nonlinear nonlocal system. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1707-1718. doi: 10.3934/cpaa.2017082

[12]

Hua Chen, Nian Liu. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 661-682. doi: 10.3934/dcds.2016.36.661

[13]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401

[14]

Jason R. Morris. A Sobolev space approach for global solutions to certain semi-linear heat equations in bounded domains. Conference Publications, 2009, 2009 (Special) : 574-582. doi: 10.3934/proc.2009.2009.574

[15]

Henri Schurz. Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 353-363. doi: 10.3934/dcdss.2008.1.353

[16]

Frank Pörner, Daniel Wachsmuth. Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations. Mathematical Control & Related Fields, 2018, 8 (1) : 315-335. doi: 10.3934/mcrf.2018013

[17]

Wen Feng, Milena Stanislavova, Atanas Stefanov. On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1371-1385. doi: 10.3934/cpaa.2018067

[18]

Zheng-Chao Han, YanYan Li. On the local solvability of the Nirenberg problem on $\mathbb S^2$. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 607-615. doi: 10.3934/dcds.2010.28.607

[19]

J. F. Padial. Existence and estimate of the location of the free-boundary for a non local inverse elliptic-parabolic problem arising in nuclear fusion. Conference Publications, 2011, 2011 (Special) : 1176-1185. doi: 10.3934/proc.2011.2011.1176

[20]

Enrique Fernández-Cara, Arnaud Münch. Numerical null controllability of semi-linear 1-D heat equations: Fixed point, least squares and Newton methods. Mathematical Control & Related Fields, 2012, 2 (3) : 217-246. doi: 10.3934/mcrf.2012.2.217

2016 Impact Factor: 0.781

Metrics

  • PDF downloads (0)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]