Partial Differential Equations, Nonlinear Analysis

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

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

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

[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. 52（2020），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 disc，Nonlinear Analysis: Real World Applications, 51(2020), 103008 (with Jie Wan and Guodong Wang)

[58] Existence and regularity of co-rotating and traveling-wave vortex solutions for the generalized SQG equation, Journal of Differential Equations, 299 (2021) 429–462.(with Guolin Qin, Weicheng Zhan, Changjun Zou)

[59] Rotating vortex patches for the planar Euler equations in a disk, Journal of Differential Equations, 275(2021), 509-532.(with Jie Wan, Guodong Wang, Weicheng Zhan)

[60] Desingularization of vortex rings in 3 dimensional Euler flows , Journal of Differential Equations 270 (2021) , 1258–1297.(with Jie Wan, Weicheng Zhan)

[61] Steady vortex patches near a nontrivial irrotational flow, SCIENCE CHINA Mathematics, 64(2021), 947–962.(with Guodong Wang, Weicheng Zhan)

[62] Singularly perturbed methods for nonlinear elliptic problems, Cambridge Studies in Advanced Mathematics, 191. Cambridge University Press, Cambridge, 2021. ix+252 pp. ISBN: 978-1-108-83683-8 .(with Peng Shuangjie, Yan Shusen)

[63] Super poly-harmonic properties, Liouville theorems and classification of nonnegative solutions to equations involving higher-order fractional Laplacians, Transactions of the American Mathematical Society, 374 (2021),  4781–4813. (with Dai Wei, Qin Guolin)

[64] The number of positive solutions to the Brezis - Nirenberg problem, Transactions of the American Mathematical Society, 374(2021)，1947–1985.(with Peng Luo, Shuangjie Peng)

[65] Traveling vortex pairs for 2D incompressible Euler equations, Calc. Var. & PDE., 60(2021) 190.(with Shanfa Lai, Weicheng Zhan)

[66] Standing waves with prescribed mass for the Schrödinger equations with van der Waals type potentials, J. Differential Equations, 276 (2021) 228–263.(with Huifang Jia, Xiao Luo)

[67] Multiscale steady vortex patches for 2D incompressible Euler equations, SIAM J. Math. Anal. , 54 (2022), no. 2, 1488–1514.(with Daomin Cao , Wan Jie)

[68] Daomin Cao, Yu Weilin, Zou Changjun, Local uniqueness of vortices for 2D steady Euler flow in a bounded domain, J. Funct. Anal. , 283 (2022), no. 8, Paper No. 109603, 43 pp.

[69] Daomin Cao,Qin Guolin, Zhan Weicheng, Zou Changjun, On the global classical solutions for the generalized SQG equation, J. Funct. Anal. , 283 (2022), no. 2, Paper No. 109503, 37.

[70] Daomin, Cao Wan Jie, Wang Guodong, Zhan Weicheng, Asymptotic behaviour of global vortex rings, Nonlinearity 35 (2022), no. 7, 3680–3705.

[71] Daomin Cao, Zhan Weicheng, Zou Changjun, On desingularization of steady vortex for the lake equations, IMA J. Appl. Math. 87 (2022), no. 1, 50–79.

[72] Daomin Cao, Qin Guolin, Zou Changjun, Vortex patch problem for steady lake equation, J. Math. Phys. 63 (2022), no. 6, Paper No. 061508.

[73] Daomin Cao, Yiming Su, Deng Zhang, On uniqueness of multi-bubble blow-up solutions and multi-solitons to L^2-critical nonlinear Schrödinger equations, Arch. Rational Mech. Anal. (2023) 247:4

[74] Daomin Cao, Guolin Qin, Weicheng Zhan, Changjun Zou, Global solutions for the generalized SQG equation and rearrangements, Transactions of the American Mathematical Society, 2023, 376(3): 2181-2211.

[75] Daomin Cao, Guolin Qin, Weicheng Zhan, Changjun Zou, Remarks on orbital stability of steady vortex rings, Transactions of the American Mathematical Society, 2023, 376(5): 3377-3395.

[76] Daomin Cao, Weilin Yu, Shusen Yan, Existence of multiple boundary vortex in planar incompressible Euler flows, SIAM J .Math. Anal., 55(2023), 5189-5220.

[77] Daomin Cao, Jie Wan, Helical vortices with small cross-section for 3D incompressible Euler equation, Journal of Functional Analysis, 284 (2023) 109836.

[78] Daomin Cao, Guolin Qin, Weicheng Zhan, Changjun Zou, Existence and stability of smooth traveling circular pairs for the generalized surface quasi-geostrophic equation, International Mathematics Research Notices, Vol. 2023, No. 6, pp. 4761–4804.

[79] Daomin Cao, Guolin Qin, Weicheng Zhan, Changjun Zou, Kármán vortex street for the generalized surface quasi-geostrophic equation, Calc. Var. &PDE., 62(2023) 168

[80] Daomin Cao, Shanfa Lai, Helical symmetry vortices for 3D incompressible Euler equations, J.Differential Equations, 360(2023),67-89.

[81] Daomin Cao, Guolin Qin, Weicheng Zhan, Changjun Zou, Existence of traveling asymmetric vortex pairs in an ideal fluid, J.Differential Equations, 351(2023),131-155.

[82] Daomin Cao, Guolin Qin, Changjun Zou, Existence of stationary vortex sheets for the 2D incompressible Euler equation, Canad. J. Math. 75 (2023), no. 3, 828–853.

[83] Daomin Cao, S.Lai, G.Qin, A note on the existence of stationary vortex patches for the SQG equation in bounded domain, Proc. Amer. Math. Soc., 151(2023), 4881–4891.

[84] Daomin Cao, S.Lai, G.Qin , Multi-peak solutions to the Schrödinger equations coupled with a neutral scalar field, Journal of Mathematical Physics,  64(2023)021510.

[85] Daomin Cao, J.Wan, Desingularization of 3D steady Euler equations with helical symmetry, Calc. Var.& PDE., 62(2023) 259

[86] Daomin Cao and Jie Wan, Structure of Green’s function of elliptic equations and helical vortex patches for 3D incompressible Euler equations, Mathematische Annalen,388 (2024), 2627–2669.

[87] Daomin Cao, Shanfa Lai, Guolin Qin, Slow traveling-wave solutions for the generalized surface quasi-geostrophic equation, Journal of Functional Analysis, 287 (2024) 110570.

[88] Daomin Cao, Boquan, Shanfa Lai, Concentrated solutions with helical symmetry for the 3D Euler equation and rearrangements, Journal of Differential Equations, 394 (2024) 152–173.

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

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