General

Daomin Cao, Professor of Mathematics,

Email: dmcao@amt.ac.cn

Telephone: 010-62571859

Address: 55 ZhongGuan Cun Dong Lu, Haidian District, Beijing, P.R.China

Postcode: 100190

### Research Areas

Partial Differential Equations, Nonlinear Analysis

### Education

-- Ph.D,  Academy of Mathematics and Systems Science, CAS, 1989

-- Master, Department of Mathematics, Xiangtan University, 1986

-- Bachelor, Department of Mathematics, Xiangtan University, 1983

### Publications

[1] On the existence and nodal character of solutions of semilinear elliptic equations. Acta Mathematica Scientia, 8(1988),345-395.(with Zhu Xiping)
[2] The concentration-compactness principle in nonlinear elliptic equations. Acta Mathematica Scientia, 9(1989),307-323.(with Zhu Xiping)
[3] Bifurcation for quasilinear equations on $R^N$ with  natural growth conditions. Proc. Royal Soc. Edinburgh, 113A(1989), 215-228.(with  G.Li and S.Yan)
[4] Positive solutions and bifurcation from the essential spectrum of a semilinear elliptic equation on $R^N$.
Nonlinear Analysis,TMA, 15(1990), 1045-1052.
[5] Bifurcation for quasilinear elliptic system on $R^N$ with natural growth conditions. J. London Math.Soc., 44(1991),  514-524.
[6] Nontrivial solutions of semilinear elliptic equations with critical exponent in $R^2$. Comm. Part. Diff. Equat., 17(1992), 407-435.
[7] Multiple solutions of a Neumann problem in an exterior domain. Comm. Part. Diff. Equat., 18(1993), 687-700.
[8] Multiple solutions of a semilinear elliptic equation on $R^N$. Ann. Inst. H. Poincare, Anal. Non lineaire, 10(1993), 593-604.
[9] Multiple solutions of inhomogeneous elliptic equations involving critical Sobolev exponent. Proc. Royal Soc.Edinburgh, 124A(1994),  1177-1191.(with G.Li and H.Zhou)
[10] Multiple positive and nodal solutions for semilinear elliptic equations involving critical Sobolev exponents. Indiana Univ.Math. J., 44(1995), 1249-1271.(with E.S.Noussair)
[11] On the existence of multiple solutions of nonhomogeneous equations involving critical Sobolev exponent. Z. Angew. Math. Phys, 47(1996),  89-96.(with H.Zhou)
[12] Multiplicity of positive and nodal solutions for nonlinear elliptic problems. Ann. Inst.H. Poincare, Anal. Nonlineaire, 13(1996), 557-588.(with E.S.Noussair)
[13] On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems. Discrete and Continuous Dynamical System, 2(1996), 221-236.(with E.N.Dancer,
E.S.Noussair and S.Yan)
[14] Existence and uniqueness results on single-peaked solutions of a semilinear problem. Ann. Inst. H.Poincare, Anal. Non lineaire, 15(1998), 73-111.(with E.S.Noussair and S.Yan)
[15] On the existence of multi-peaked solutions to a semilinear Neumann problem. Duke Math. J.,97(1999), 261-300. (with T.Kupper)
[16] Solutions with multiple "peaks" for nonlinear elliptic equations. Proc.Royal Soc.
Edinburgh,126A (1999), 235 - 264.(with E.S.Noussair and S.Yan)
[17] Multi-peak solutions for a singularly perturbed semilinear elliptic problem. J.Diff. Equat., 166 (2000), 266 -289.(with E.S.Noussair)
[18] Existence of symmetric multi- peaked solutions to singularly  perturbed semilinear problems. Comm. Part. Diff. Equat., 25(2000), 2185-2232.(with E.S.Noussair)
[19] Existence and nonexistence of interior-peaked solution for a nonlinear Neumann problem. Pacific J.Math.,200(2001), 19-40.(with E.S.Noussair and S.Yan)
[20] A Neumann problem in exterior domain, Manuscripta Mathematica.106(2001), 63-74.(with M. Lucia and H.S. Zhou)
[21] On the scalar curvature equation $-∆ u=(1+εK)u^{\frac{N+2}{N-2}}$ in $R^N$ . Calc. Var. and PDE, 15(2002), 403-419.(with E.S.Noussair and S.Yan)
[22] Uniqueness of positive multi-lump bound states of nonlinear Schrodinger equations. Math. Zeit., 243(2003), 599-642.(with H.P. Heinz)
[23] A compactness result for singular elliptic problems involving critical Sobolev exponent. Proc. Amer.Math. Soc.,131(2003), 1857 - 1866.(with S.Peng)
[24] Solutions for semilinear elliptic equations with critical exponents and Hardy potential. J. Diff. Equat., 205(2004), 521--537.(with P.Han)
[25] Multi-bump standing waves with a critical frequency for nonlinear Schrodinger equations. J. Diff. Equat., 203(2004), 292--312.(with E.S.Noussair)
[26] High energy positive solutions for Neumann problem for an elliptic system of equations with critical nonlinearity. Calc. Var. and PDE, 25(2005), 161-185. (with P. Han)
[27] Asymptotic behavior for elliptic problems with singular coefficient and nearly critical Sobolev growth. Annali di Mat. Pura ed Appl., 185 (2006), 189-205. (with S.Peng)
[28] Multi-bump bound states of Schrodinger equations with a critical frequency. Math. Ann., 336(2006), 925-948.(with S.Peng)
[29] Effective macroscopic dynamics of stochastic partial differential equations in perforated domains. SIAM J.Math.Anal., 38 (2007), 1508 -1527. (with J. Duan and W.Wang)
[30] Multiscale-bump standing waves with a critical frequency for nonlinear Schrodinger equations. Trans. Amer. Math. Soc.,360 (2008), 3813-3837.(with E.S.Noussair and S.Yan)
[31] Semi-classical bound states for Schrodinger equations with potentials vanishing or unbounded at infinity. Comm. Part. Diff. Equat., 34 (2009), 1566--1591.(with S.Peng)
[32] Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential. Calc. Var. and  PDE., 38(2010), 471-501.(with S.Yan)
[33] Multiplicity of solutions for the plasma problem in two dimensions. Adv. Math., 225 (2010), 2741-2785.(with S.Peng and S.Yan)
[34] Local exact controllability of the Navier-Stokes equations with the condition on the pressure on parts of the boundary. SIAM J. Control Optim., 48(2010), 3805-3837.(with T.Kim)
[35] Divergent solutions to the 5D Hartree equations. Colloq. Math., 125 (2011), 255-287.(with Q.Guo)
[36] Infinitely many solutions for p-Laplacian equation involving critical Sobolev growth. J. Funct. Anal., 262 (2012), 2861-2902.(with S.Peng and S.Yan)
[37] Continuous dependence of Cauchy problem for nonlinear Schrodinger equation in $H^s$. J. Diff. Equat., 255 (2013),2018-2064.(with W.Dai and W.Yang)
[38] Concentration of solutions for the Yamabe problem on half -spaces.  Proc. Roy. Soc. Edinburgh, Sect. A 143 (2013),no. 1, 73-99.(with S.Peng)
[39] On the Webster scalar curvature problem on the CR sphere with a cylindrical- type symmetry. J. Geom.  Anal., 23(2013),1674-1702. (with S.Peng and S.Yan)
[40] Minimal blow-up solutions of mass-critical inhomogeneous Hartree equation. J. Math. Phys.,
54 (2013), no. 12, 121511, 25 pp. (with Su Yiming)
[41] Regularization of point vortices pairs for the Euler equation in dimension two. Arch. Ration. Mech. Anal., 212 (2014), no. 1, 179–217. (with Liu Zhongyuan and Wei Juncheng)
[42] Uniqueness of positive bound states with multi-bump for nonlinear Schrödinger equations. Calc. Var. Partial Differential Equations, 54 (2015), no. 4, 4037–4063. (with  Li Shuanglong and Luo Peng)
[43] Dynamics for a stochastic reaction-diffusion equation with additive noise. J. Differential Equations, 259 (2015), no. 3, 838–872. (with Sun Chunyou and Yang Meihua)
[44] Planar vortex patch problem in incompressible steady flow. Adv. Math., 270 (2015), 263–301.
(with  Peng Shuangjie and Yan Shusen)
[45] Non-stationary Navier-Stokes Equations with Mixed Boundary Conditions. J. Math. Sci. Univ. Tokyo, 24 (2017), 159–194.(with Tujin Kim)
[46] Nodal solutions for a supercritical semilinear problem with variable exponent. Calc. Var. Partial Differential Equations, 57 (2018), (with Li Shuanglong and Liu Zhongyuan)
[47]  Sign-changing bubble tower solutions for the supercritical Hénon-type equations. Ann. Mat. Pura. Appl. ,197(2018),1227–1246.(with Zhongyuan Liu and Shuangjie Peng)
[48] Nodal solutions for a supercritical semilinear problem with variable exponent. Calculus of Variations and Partial Differential Equations, 57(2018), Art. 38, 19 pp. (with Zhongyuan Liu and Shuanglong Li)
[49] Steady vortex patches with opposite rotation directions in a planar ideal fluid. Calculus of Variations and Partial Differential Equations, 58(2019) 58:75 （with Guodong Wang）
[50] Steady vortex patch solutions to the vortex-wave system. Nonlinearity，32(2019),1182-1904（with Guodong Wang）
[51] Nonlinear orbital stability for planar vortex patches. Proceedings of the American Mathematical Society,147(2019),775-784.(with Jie Wan and Guodong Wang)
[52] Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity.
Proceedings of the Royal Society of Edinburgh, 149(2019),979-994. (with Wei Dai)

[53] Local uniqueness for vortex patch problem in incompressible planar steady flow. Journal de Mathématiques Pures et Appliquées,131(2019), 251-289.(with Yuxia Guo, Shuangjie Peng and Shusen Yan)

[54] A note on steady vortex flows in two dimensions, Proceedings of the American Mathematical Society,148 (2020), 1153 -1159.with Guodong Wang

[55] Desingularization of vortices for 2D steady Euler flows via the vorticity method, SIAM J. Math., Anal. 522020），5363-5388 with G. Wang and W. Zhan

[56] Existence of Steady Multiple Vortex Patches to the Vortex-wave System, Pacific Math. J.,308(2020), 257-279. with Guodong Wang

[57] Steady vortex patch with polygonal symmetry for the planar Euler equations in a discNonlinear Analysis: Real World Applications, 51(2020), 103008 (with Jie Wan and Guodong Wang)

### Research Interests

Partial Differential Equations of Elliptic Type，Equations Arising from Fluid, Calculus of Variations

### Students

韩丕功  博士研究生  070104-应用数学 (Pigong      Han)

唐仲伟  博士研究生  070104-应用数学  (Zhongwei Tang)

赖善发  博士研究生  070104-应用数学  (Shanfa Lai)