American Institute of Mathematical Sciences

Advanced
January  2018, 17(1): 209-230. doi: 10.3934/cpaa.2018013

Infinitely many blowing-up solutions for Yamabe-type problems on manifolds with boundary

Shaodong Wang

McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montreal, Quebec H3A 0B9, Canada

Received  April 2017 Revised  June 2017 Published  September 2017

Fund Project: The author is supported by the China Scholarship Council.

We consider the Yamabe-type problem
$\begin{equation*} \begin{cases} Δ_g u+fu=0 \ in\ M\\ \frac{\partial u}{\partial ν}+hu=u^{\frac{n}{n-2}}\ on\ \partial M \end{cases}\end{equation*} $
when
$(M,g)$
is the standard half sphere of dimensions
$n≥ 3$
. We establish existence results of positive blowing-up solutions with unbounded energy to this problem for all dimensions
$n≥ 3$
.
Keywords: Blow-up solutions, Yamabe-type problem, half space, Neumann problem, half sphere.
Mathematics Subject Classification: Primary:35J25, 35J65;Secondary:53A30.
Citation: Shaodong Wang. Infinitely many blowing-up solutions for Yamabe-type problems on manifolds with boundary. Communications on Pure & Applied Analysis, 2018, 17 (1) : 209-230. doi: 10.3934/cpaa.2018013
References:
[1]

S. Almaraz, An existence theorem of conformal scalar-flat metrics on manifolds with boundary, Pacific J. Math., 248 (2010), 1-22.  doi: 10.2140/pjm.2010.248.1.  Google Scholar

[2]

S. Almaraz, A compactness theorem for scalar-flat metrics on manifolds with boundary, Calc. Var. Partial Differential Equations, 41 (2011), 341-386.  doi: 10.1007/s00526-010-0365-8.  Google Scholar

[3]

S. Almaraz, Blow-up phenomena for scalar-flat metrics on manifolds with boundary, J. Differential Equations, 251 (2011), 1813-1840.  doi: 10.1016/j.jde.2011.04.013.  Google Scholar

[4]

S. Brendle, Blow-up phenomena for the Yamabe equation, J. Amer. Math. Soc., 21 (2008), 951-979.  doi: 10.1090/S0894-0347-07-00575-9.  Google Scholar

[5]

S. Brendle and S. Chen, An existence theorem for the Yamabe problem on manifolds with boundary, J. Eur. Math. Soc. (JEMS), 16 (2014), 991-1016.  doi: 10.4171/JEMS/453.  Google Scholar

[6]

S. Brendle and F. Marques, Blow-up phenomena for the Yamabe equation. Ⅱ, J. Differential Geom., 81 (2009), 225-250.   Google Scholar

[7]

S. Chen, Conformal deformation to scalar flat metrics with constant mean curvature on the boundary in higher dimensions, preprint, arXiv: 0912.1302. Google Scholar

[8]

W. Chen, J. Wei and S. Yan, Infinitely many solutions for the Schrödinger equations in $\mathbb{R}^N$ with critical growth, J. Differential Equations, 252 (2012), 2425-2447.  doi: 10.1016/j.jde.2011.09.032.  Google Scholar

[9]

P. Cherrier, Problémes de Neumann non linéaires sur les variétés riemanniennes, J. Funct. Anal., 57 (1984), 154-206.  doi: 10.1016/0022-1236(84)90094-6.  Google Scholar

[10]

O. Druet, From one bubble to several bubbles: the low-dimensional case, J. Differential Geom., 63 (2003), 399-473.   Google Scholar

[11]

O. Druet, Compactness for Yamabe metrics in low dimensions, Int. Math. Res. Not., (2004), 1143-1191.  doi: 10.1155/S1073792804133278.  Google Scholar

[12]

O. Druet and E. Hebey, Blow-up examples for second order elliptic PDEs of critical Sobolev growth, Trans. Amer. Math. Soc., 357 (2005), 1915-1929.  doi: 10.1090/S0002-9947-04-03681-5.  Google Scholar

[13]

M. Disconzi and M. Khuri, Compactness and non-compactness for the Yamabe problem on manifolds with boundary, J. Reine Angew. Math., 724 (2017), 145-201.  doi: 10.1515/crelle-2014-0083.  Google Scholar

[14]

J. Escobar, Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary, Ann. of Math. (2), 136 (1992), 1-50.  doi: 10.2307/2946545.  Google Scholar

[15]

J. Escobar, The Yamabe problem on manifolds with boundary, J. Differential Geom., 35 (1992), 21-84.   Google Scholar

[16]

P. Esposito, A. Pistoia and J. Vétois, The effect of linear perturbations on the Yamabe problem, Math. Ann., 358 (2014), 511-560.  doi: 10.1007/s00208-013-0971-9.  Google Scholar

[17]

V. Felli and M. Ahmedou, Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries, Math. Z., 244 (2003), 175-210.  doi: 10.1007/s00209-002-0486-7.  Google Scholar

[18]

V. Felli and M. Ahmedou, A geometric equation with critical nonlinearity on the boundary, Pacific J. Math., 218 (2005), 75-99.  doi: 10.2140/pjm.2005.218.75.  Google Scholar

[19]

M. Ghimenti, A. Micheletti and A. Pistoia, On Yamabe-type problems on Riemannian manifolds with boundary, Pacific J. Math., 284 (2016), 79-102.  doi: 10.2140/pjm.2016.284.79.  Google Scholar

[20]

M. Ghimenti, A. Micheletti and A. Pistoia, Linear Perturbations of the Yamabe problem on manifolds with boundary preprint, arXiv: 1611.01336. doi: 10.2140/pjm.2016.284.79.  Google Scholar

[21]

Z. Han and Y. Li, The Yamabe problem on manifolds with boundary: existence and compactness results, Duke Math. J., 99 (1999), 489-542.  doi: 10.1215/S0012-7094-99-09916-7.  Google Scholar

[22]

E. Hebey, Compactness and Stability for Nonlinear Elliptic Equations Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2014. doi: 10.4171/134.  Google Scholar

[23]

E. Hebey and J. Wei, Resonant states for the static Klein-Gordon-Maxwell-Proca system, Math. Res. Lett., 19 (2012), 953-967.  doi: 10.4310/MRL.2012.v19.n4.a18.  Google Scholar

[24]

M. Khuri, F. Marques and R. Schoen, A compactness theorem for the Yamabe problem, J. Differential Geom., 81 (2009), 143-196.   Google Scholar

[25]

Y. Li and L. Zhang, A Harnack type inequality for the Yamabe equation in low dimensions, Calc. Var. Partial Differential Equations, 20 (2004), 133-151.  doi: 10.1007/s00526-003-0224-y.  Google Scholar

[26]

Y. Li and L. Zhang, Compactness of solutions to the Yamabe problem. Ⅱ, Calc. Var. Partial Differential Equations, 24 (2005), 185-237.  doi: 10.1007/s00526-004-0320-7.  Google Scholar

[27]

Y. Li and L. Zhang, Compactness of solutions to the Yamabe problem. Ⅲ, J. Funct. Anal., 245 (2007), 438-474.  doi: 10.1016/j.jfa.2006.11.010.  Google Scholar

[28]

Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417.  doi: 10.1215/S0012-7094-95-08016-8.  Google Scholar

[29]

Y. Li and M. Zhu, Yamabe type equations on three-dimensional Riemannian manifolds, Commun. Contemp. Math., 1 (1999), 1-50.  doi: 10.1142/S021919979900002X.  Google Scholar

[30]

F. Marques, A priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Differential Geom., 71 (2005), 315-346.   Google Scholar

[31]

F. Marques, Existence results for the Yamabe problem on manifolds with boundary, Indiana Univ. Math. J., 54 (2005), 1599-1620.  doi: 10.1512/iumj.2005.54.2590.  Google Scholar

[32]

F. Marques, Conformal deformations to scalar-flat metrics with constant mean curvature on the boundary, Comm. Anal. Geom., 15 (2007), 381-405.   Google Scholar

[33]

R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., 20 (1984), 479-495.   Google Scholar

[34]

R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, in Lecture Notes in Math., Springer, Berlin, (1989), 120-154. doi: 10.1007/BFb0089180.  Google Scholar

[35]

R. Schoen, On the number of constant scalar curvature metrics in a conformal class, in Pitman Monogr. Surveys Pure Appl. Math., Longman Sci. Tech., Harlow, (1991), 311-320.  Google Scholar

[36]

P. Thizy and J. Vétois, Positive clusters for smooth perturbations of a critical elliptic equation in dimension four and five, preprint, arXiv: 1603.06479. Google Scholar

[37]

N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 265-274.   Google Scholar

[38]

J. Vétois and S. Wang, Infinitely many solutions for cubic nonlinear Schrödinger equations in dimension four, preprint 2017. Google Scholar

[39]

L. Wang, J. Wei and S. Yan, A Neumann problem with critical exponent in nonconvex domains and Lin-Ni's conjecture, Trans. Amer. Math. Soc., 362 (2010), 4581-4615.  doi: 10.1090/S0002-9947-10-04955-X.  Google Scholar

[40]

L. Wang, J. Wei and S. Yan, On Lin-Ni's conjecture in convex domains, Proc. Lond. Math. Soc. (3), 102 (2011), 1099-1126.  doi: 10.1112/plms/pdq051.  Google Scholar

[41]

J. Wei and S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $\mathbb R^N$, Calc. Var. Partial Differential Equations, 37 (2010), 423-439.  doi: 10.1007/s00526-009-0270-1.  Google Scholar

[42]

J. Wei and S. Yan, Infinitely many solutions for the prescribed scalar curvature problem on $\mathbb S^N$, J. Funct. Anal., 258 (2010), 3048-3081.  doi: 10.1016/j.jfa.2009.12.008.  Google Scholar

[43]

J. Wei and S. Yan, On a stronger Lazer-McKenna conjecture for Ambrosetti-Prodi type problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 9 (2010), 423-457.   Google Scholar

[44]

J. Wei and S. Yan, Infinitely many positive solutions for an elliptic problem with critical or supercritical growth, J. Math. Pures Appl. (9), 96 (2011), 307-333.  doi: 10.1016/j.matpur.2011.01.006.  Google Scholar

show all references

References:
[1]

S. Almaraz, An existence theorem of conformal scalar-flat metrics on manifolds with boundary, Pacific J. Math., 248 (2010), 1-22.  doi: 10.2140/pjm.2010.248.1.  Google Scholar

[2]

S. Almaraz, A compactness theorem for scalar-flat metrics on manifolds with boundary, Calc. Var. Partial Differential Equations, 41 (2011), 341-386.  doi: 10.1007/s00526-010-0365-8.  Google Scholar

[3]

S. Almaraz, Blow-up phenomena for scalar-flat metrics on manifolds with boundary, J. Differential Equations, 251 (2011), 1813-1840.  doi: 10.1016/j.jde.2011.04.013.  Google Scholar

[4]

S. Brendle, Blow-up phenomena for the Yamabe equation, J. Amer. Math. Soc., 21 (2008), 951-979.  doi: 10.1090/S0894-0347-07-00575-9.  Google Scholar

[5]

S. Brendle and S. Chen, An existence theorem for the Yamabe problem on manifolds with boundary, J. Eur. Math. Soc. (JEMS), 16 (2014), 991-1016.  doi: 10.4171/JEMS/453.  Google Scholar

[6]

S. Brendle and F. Marques, Blow-up phenomena for the Yamabe equation. Ⅱ, J. Differential Geom., 81 (2009), 225-250.   Google Scholar

[7]

S. Chen, Conformal deformation to scalar flat metrics with constant mean curvature on the boundary in higher dimensions, preprint, arXiv: 0912.1302. Google Scholar

[8]

W. Chen, J. Wei and S. Yan, Infinitely many solutions for the Schrödinger equations in $\mathbb{R}^N$ with critical growth, J. Differential Equations, 252 (2012), 2425-2447.  doi: 10.1016/j.jde.2011.09.032.  Google Scholar

[9]

P. Cherrier, Problémes de Neumann non linéaires sur les variétés riemanniennes, J. Funct. Anal., 57 (1984), 154-206.  doi: 10.1016/0022-1236(84)90094-6.  Google Scholar

[10]

O. Druet, From one bubble to several bubbles: the low-dimensional case, J. Differential Geom., 63 (2003), 399-473.   Google Scholar

[11]

O. Druet, Compactness for Yamabe metrics in low dimensions, Int. Math. Res. Not., (2004), 1143-1191.  doi: 10.1155/S1073792804133278.  Google Scholar

[12]

O. Druet and E. Hebey, Blow-up examples for second order elliptic PDEs of critical Sobolev growth, Trans. Amer. Math. Soc., 357 (2005), 1915-1929.  doi: 10.1090/S0002-9947-04-03681-5.  Google Scholar

[13]

M. Disconzi and M. Khuri, Compactness and non-compactness for the Yamabe problem on manifolds with boundary, J. Reine Angew. Math., 724 (2017), 145-201.  doi: 10.1515/crelle-2014-0083.  Google Scholar

[14]

J. Escobar, Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary, Ann. of Math. (2), 136 (1992), 1-50.  doi: 10.2307/2946545.  Google Scholar

[15]

J. Escobar, The Yamabe problem on manifolds with boundary, J. Differential Geom., 35 (1992), 21-84.   Google Scholar

[16]

P. Esposito, A. Pistoia and J. Vétois, The effect of linear perturbations on the Yamabe problem, Math. Ann., 358 (2014), 511-560.  doi: 10.1007/s00208-013-0971-9.  Google Scholar

[17]

V. Felli and M. Ahmedou, Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries, Math. Z., 244 (2003), 175-210.  doi: 10.1007/s00209-002-0486-7.  Google Scholar

[18]

V. Felli and M. Ahmedou, A geometric equation with critical nonlinearity on the boundary, Pacific J. Math., 218 (2005), 75-99.  doi: 10.2140/pjm.2005.218.75.  Google Scholar

[19]

M. Ghimenti, A. Micheletti and A. Pistoia, On Yamabe-type problems on Riemannian manifolds with boundary, Pacific J. Math., 284 (2016), 79-102.  doi: 10.2140/pjm.2016.284.79.  Google Scholar

[20]

M. Ghimenti, A. Micheletti and A. Pistoia, Linear Perturbations of the Yamabe problem on manifolds with boundary preprint, arXiv: 1611.01336. doi: 10.2140/pjm.2016.284.79.  Google Scholar

[21]

Z. Han and Y. Li, The Yamabe problem on manifolds with boundary: existence and compactness results, Duke Math. J., 99 (1999), 489-542.  doi: 10.1215/S0012-7094-99-09916-7.  Google Scholar

[22]

E. Hebey, Compactness and Stability for Nonlinear Elliptic Equations Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2014. doi: 10.4171/134.  Google Scholar

[23]

E. Hebey and J. Wei, Resonant states for the static Klein-Gordon-Maxwell-Proca system, Math. Res. Lett., 19 (2012), 953-967.  doi: 10.4310/MRL.2012.v19.n4.a18.  Google Scholar

[24]

M. Khuri, F. Marques and R. Schoen, A compactness theorem for the Yamabe problem, J. Differential Geom., 81 (2009), 143-196.   Google Scholar

[25]

Y. Li and L. Zhang, A Harnack type inequality for the Yamabe equation in low dimensions, Calc. Var. Partial Differential Equations, 20 (2004), 133-151.  doi: 10.1007/s00526-003-0224-y.  Google Scholar

[26]

Y. Li and L. Zhang, Compactness of solutions to the Yamabe problem. Ⅱ, Calc. Var. Partial Differential Equations, 24 (2005), 185-237.  doi: 10.1007/s00526-004-0320-7.  Google Scholar

[27]

Y. Li and L. Zhang, Compactness of solutions to the Yamabe problem. Ⅲ, J. Funct. Anal., 245 (2007), 438-474.  doi: 10.1016/j.jfa.2006.11.010.  Google Scholar

[28]

Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417.  doi: 10.1215/S0012-7094-95-08016-8.  Google Scholar

[29]

Y. Li and M. Zhu, Yamabe type equations on three-dimensional Riemannian manifolds, Commun. Contemp. Math., 1 (1999), 1-50.  doi: 10.1142/S021919979900002X.  Google Scholar

[30]

F. Marques, A priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Differential Geom., 71 (2005), 315-346.   Google Scholar

[31]

F. Marques, Existence results for the Yamabe problem on manifolds with boundary, Indiana Univ. Math. J., 54 (2005), 1599-1620.  doi: 10.1512/iumj.2005.54.2590.  Google Scholar

[32]

F. Marques, Conformal deformations to scalar-flat metrics with constant mean curvature on the boundary, Comm. Anal. Geom., 15 (2007), 381-405.   Google Scholar

[33]

R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., 20 (1984), 479-495.   Google Scholar

[34]

R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, in Lecture Notes in Math., Springer, Berlin, (1989), 120-154. doi: 10.1007/BFb0089180.  Google Scholar

[35]

R. Schoen, On the number of constant scalar curvature metrics in a conformal class, in Pitman Monogr. Surveys Pure Appl. Math., Longman Sci. Tech., Harlow, (1991), 311-320.  Google Scholar

[36]

P. Thizy and J. Vétois, Positive clusters for smooth perturbations of a critical elliptic equation in dimension four and five, preprint, arXiv: 1603.06479. Google Scholar

[37]

N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 265-274.   Google Scholar

[38]

J. Vétois and S. Wang, Infinitely many solutions for cubic nonlinear Schrödinger equations in dimension four, preprint 2017. Google Scholar

[39]

L. Wang, J. Wei and S. Yan, A Neumann problem with critical exponent in nonconvex domains and Lin-Ni's conjecture, Trans. Amer. Math. Soc., 362 (2010), 4581-4615.  doi: 10.1090/S0002-9947-10-04955-X.  Google Scholar

[40]

L. Wang, J. Wei and S. Yan, On Lin-Ni's conjecture in convex domains, Proc. Lond. Math. Soc. (3), 102 (2011), 1099-1126.  doi: 10.1112/plms/pdq051.  Google Scholar

[41]

J. Wei and S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $\mathbb R^N$, Calc. Var. Partial Differential Equations, 37 (2010), 423-439.  doi: 10.1007/s00526-009-0270-1.  Google Scholar

[42]

J. Wei and S. Yan, Infinitely many solutions for the prescribed scalar curvature problem on $\mathbb S^N$, J. Funct. Anal., 258 (2010), 3048-3081.  doi: 10.1016/j.jfa.2009.12.008.  Google Scholar

[43]

J. Wei and S. Yan, On a stronger Lazer-McKenna conjecture for Ambrosetti-Prodi type problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 9 (2010), 423-457.   Google Scholar

[44]

J. Wei and S. Yan, Infinitely many positive solutions for an elliptic problem with critical or supercritical growth, J. Math. Pures Appl. (9), 96 (2011), 307-333.  doi: 10.1016/j.matpur.2011.01.006.  Google Scholar

[1]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259
[2]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264
[3]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453
[4]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216
[5]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253
[6]

Tahir Aliyev Azeroğlu, Bülent Nafi Örnek, Timur Düzenli. Some results on the behaviour of transfer functions at the right half plane. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020106
[7]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171
[8]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354
[9]

Adrian Constantin, Darren G. Crowdy, Vikas S. Krishnamurthy, Miles H. Wheeler. Stuart-type polar vortices on a rotating sphere. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 201-215. doi: 10.3934/dcds.2020263
[10]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052
[11]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251
[12]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364
[13]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384
[14]

Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108
[15]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031
[16]

Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075
[17]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261
[18]

Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099
[19]

Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102
[20]

Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (63)
  • HTML views (150)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences