• Previous Article
    Quantization conditions of eigenvalues for semiclassical Zakharov-Shabat systems on the circle
  • DCDS Home
  • This Issue
  • Next Article
    The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form
August  2018, 38(8): 3831-3850. doi: 10.3934/dcds.2018166

Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros

1. 

Departamento de Matemática, Universidad Técnica Federico Santa María, Avenida España 1680, Casilla 110-V, Valparaíso, Chile

2. 

Departamento de Matemática, Instituto de Ciêencias Matemáticas e de Computação, Universidade de São Paulo - Campus de São Carlos, Caixa Postal 668, 13560-970, São Carlos SP, Brazil

* Corresponding author: E. Massa

Received  August 2017 Revised  April 2018 Published  May 2018

Fund Project: The first author gratefully acknowledges financial support from Fondecyt grants 1161635, 1171532 and 1171691
The author E. Massa was supported by: grant #2014/25398-0, São Paulo Research Foundation (FAPESP) and grants #308354/2014-1, #303447/2017-6, CNPq/Brazil.

In this paper we consider the equation $(-Δ)^k\, u = λ f(x, u)+μ g(x, u)$ with Navier boundary conditions, in a bounded and smooth domain. The main interest is when the nonlinearity is nonnegative but admits a zero and $f, g$ are, respectively, identically zero above and below the zero. We prove the existence of multiple positive solutions when the parameters lie in a region of the form $λ>\overline λ$ and $0 < μ< \overlineμ(λ)$, then we provide further conditions under which, respectively, the bound $\overlineμ(λ)$ is either necessary, or can be removed.

Citation: Leonelo Iturriaga, Eugenio Massa. Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3831-3850. doi: 10.3934/dcds.2018166
References:
[1]

A. AmbrosettiH. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.  Google Scholar

[2]

T. Bartsch and Z. Liu, Multiple sign changing solutions of a quasilinear elliptic eigenvalue problem involving the p-Laplacian, Commun. Contemp. Math., 6 (2004), 245-258.  doi: 10.1142/S0219199704001306.  Google Scholar

[3]

F. BernisJ. García Azorero and I. Peral, Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order, Adv. Differential Equations, 1 (1996), 219-240.   Google Scholar

[4]

H. Brezis and L. Nirenberg, H1 versus C1 local minimizers, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 465-472.   Google Scholar

[5]

D. G. de FigueiredoJ.-P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467.  doi: 10.1016/S0022-1236(02)00060-5.  Google Scholar

[6]

D. G. de FigueiredoJ.-P. Gossez and P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity, J. Eur. Math. Soc. (JEMS), 8 (2006), 269-286.  doi: 10.4171/JEMS/52.  Google Scholar

[7]

D. G. de FigueiredoP.-L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. (9), 61 (1982), 41-63.   Google Scholar

[8]

F. O. de Paiva and E. Massa, Multiple solutions for some elliptic equations with a nonlinearity concave at the origin, Nonlinear Anal., 66 (2007), 2940-2946.  doi: 10.1016/j.na.2006.04.015.  Google Scholar

[9]

J. Díaz and J. Saá, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 521-524.   Google Scholar

[10]

F. Ebobisse and M. Ahmedou, On a nonlinear fourth-order elliptic equation involving the critical Sobolev exponent, Nonlinear Anal., 52 (2003), 1535-1552.  doi: 10.1016/S0362-546X(02)00273-0.  Google Scholar

[11]

D. E. EdmundsD. Fortunato and E. Jannelli, Critical exponents, critical dimensions and the biharmonic operator, Arch. Rational Mech. Anal., 112 (1990), 269-289.  doi: 10.1007/BF00381236.  Google Scholar

[12]

J. García-Melián and L. Iturriaga, Multiplicity of solutions for some semilinear problems involving nonlinearities with zeros, Israel J. Math., 210 (2015), 233-244.  doi: 10.1007/s11856-015-1251-z.  Google Scholar

[13]

J. García-Melián and J. Sabina de Lis, Stationary profiles of degenerate problems when a parameter is large, Differential Integral Equations, 13 (2000), 1201-1232.   Google Scholar

[14]

F. GazzolaH.-C. Grunau and M. Squassina, Existence and nonexistence results for critical growth biharmonic elliptic equations, Calc. Var. Partial Differential Equations, 18 (2003), 117-143.  doi: 10.1007/s00526-002-0182-9.  Google Scholar

[15]

F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, vol. 1991 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010, Positivity preserving and nonlinear higher order elliptic equations in bounded domains. doi: 10.1007/978-3-642-12245-3.  Google Scholar

[16]

M. Guedda and L. Véron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal., 13 (1989), 879-902.  doi: 10.1016/0362-546X(89)90020-5.  Google Scholar

[17]

P. Hess, On multiple positive solutions of nonlinear elliptic eigenvalue problems, Comm. Partial Differential Equations, 6 (1981), 951-961.  doi: 10.1080/03605308108820200.  Google Scholar

[18]

J. Hulshof and R. van der Vorst, Differential systems with strongly indefinite variational structure, J. Funct. Anal., 114 (1993), 32-58.  doi: 10.1006/jfan.1993.1062.  Google Scholar

[19]

L. IturriagaS. Lorca and E. Massa, Positive solutions for the p-Laplacian involving critical and supercritical nonlinearities with zeros, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 763-771.  doi: 10.1016/j.anihpc.2009.11.003.  Google Scholar

[20]

L. IturriagaS. Lorca and E. Massa, Multiple positive solutions for the m-Laplacian and a nonlinearity with many zeros, Differential Integral Equations, 30 (2017), 145-159.   Google Scholar

[21]

L. IturriagaE. MassaJ. Sánchez and P. Ubilla, Positive solutions of the p-Laplacian involving a superlinear nonlinearity with zeros, J. Differential Equations, 248 (2010), 309-327.  doi: 10.1016/j.jde.2009.08.008.  Google Scholar

[22]

L. IturriagaE. MassaJ. Sanchez and P. Ubilla, Positive solutions for an elliptic equation in an annulus with a superlinear nonlinearity with zeros, Math. Nach., 287 (2014), 1131-1141.  doi: 10.1002/mana.201100285.  Google Scholar

[23]

A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev., 32 (1990), 537-578.  doi: 10.1137/1032120.  Google Scholar

[24]

A. C. Lazer and P. J. McKenna, Global bifurcation and a theorem of Tarantello, J. Math. Anal. Appl., 181 (1994), 648-655.  doi: 10.1006/jmaa.1994.1049.  Google Scholar

[25]

P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467.  doi: 10.1137/1024101.  Google Scholar

[26]

Z. Liu, Positive solutions of superlinear elliptic equations, J. Funct. Anal., 167 (1999), 370-398.  doi: 10.1006/jfan.1999.3446.  Google Scholar

[27]

Z. Liu, Positive solutions of a class of nonlinear elliptic eigenvalue problems, Math. Z., 242 (2002), 663-686.  doi: 10.1007/s002090100373.  Google Scholar

[28]

A. M. Micheletti and A. Pistoia, Multiplicity results for a fourth-order semilinear elliptic problem, Nonlinear Anal., 31 (1998), 895-908.  doi: 10.1016/S0362-546X(97)00446-X.  Google Scholar

[29]

A. M. Micheletti and A. Pistoia, Nontrivial solutions for some fourth order semilinear elliptic problems, Nonlinear Anal., 34 (1998), 509-523.  doi: 10.1016/S0362-546X(97)00596-8.  Google Scholar

[30]

E. S. NoussairC. A. Swanson and J. Yang, Critical semilinear biharmonic equations in RN, Proc. Roy. Soc. Edinburgh Sect. A, 121 (1992), 139-148.  doi: 10.1017/S0308210500014189.  Google Scholar

[31]

L. A. Peletier and R. van der Vorst, Existence and nonexistence of positive solutions of nonlinear elliptic systems and the biharmonic equation, Differential Integral Equations, 5 (1992), 747-767.   Google Scholar

[32]

P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl. (9), 69 (1990), 55-83.   Google Scholar

[33]

S. Takeuchi, Coincidence sets in semilinear elliptic problems of logistic type, Differential Integral Equations, 20 (2007), 1075-1080.   Google Scholar

[34]

S. Takeuchi, Partial flat core properties associated to the p-Laplace operator, Discrete Contin. Dyn. Syst., (2007), 965-973.   Google Scholar

[35]

G. Tarantello, A note on a semilinear elliptic problem, Differential Integral Equations, 5 (1992), 561-565.   Google Scholar

[36]

G. Xu and J. Zhang, Existence results for some fourth-order nonlinear elliptic problems of local superlinearity and sublinearity, J. Math. Anal. Appl., 281 (2003), 633-640.  doi: 10.1016/S0022-247X(03)00170-7.  Google Scholar

[37]

J. Zhang and Z. Wei, Multiple solutions for a class of biharmonic equations with a nonlinearity concave at the origin, J. Math. Anal. Appl., 383 (2011), 291-306.  doi: 10.1016/j.jmaa.2011.05.030.  Google Scholar

[38]

Y. Zhang, Positive solutions of semilinear biharmonic equations with critical Sobolev exponents, Nonlinear Anal., 75 (2012), 55-67.  doi: 10.1016/j.na.2011.07.065.  Google Scholar

show all references

References:
[1]

A. AmbrosettiH. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.  Google Scholar

[2]

T. Bartsch and Z. Liu, Multiple sign changing solutions of a quasilinear elliptic eigenvalue problem involving the p-Laplacian, Commun. Contemp. Math., 6 (2004), 245-258.  doi: 10.1142/S0219199704001306.  Google Scholar

[3]

F. BernisJ. García Azorero and I. Peral, Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order, Adv. Differential Equations, 1 (1996), 219-240.   Google Scholar

[4]

H. Brezis and L. Nirenberg, H1 versus C1 local minimizers, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 465-472.   Google Scholar

[5]

D. G. de FigueiredoJ.-P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467.  doi: 10.1016/S0022-1236(02)00060-5.  Google Scholar

[6]

D. G. de FigueiredoJ.-P. Gossez and P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity, J. Eur. Math. Soc. (JEMS), 8 (2006), 269-286.  doi: 10.4171/JEMS/52.  Google Scholar

[7]

D. G. de FigueiredoP.-L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. (9), 61 (1982), 41-63.   Google Scholar

[8]

F. O. de Paiva and E. Massa, Multiple solutions for some elliptic equations with a nonlinearity concave at the origin, Nonlinear Anal., 66 (2007), 2940-2946.  doi: 10.1016/j.na.2006.04.015.  Google Scholar

[9]

J. Díaz and J. Saá, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 521-524.   Google Scholar

[10]

F. Ebobisse and M. Ahmedou, On a nonlinear fourth-order elliptic equation involving the critical Sobolev exponent, Nonlinear Anal., 52 (2003), 1535-1552.  doi: 10.1016/S0362-546X(02)00273-0.  Google Scholar

[11]

D. E. EdmundsD. Fortunato and E. Jannelli, Critical exponents, critical dimensions and the biharmonic operator, Arch. Rational Mech. Anal., 112 (1990), 269-289.  doi: 10.1007/BF00381236.  Google Scholar

[12]

J. García-Melián and L. Iturriaga, Multiplicity of solutions for some semilinear problems involving nonlinearities with zeros, Israel J. Math., 210 (2015), 233-244.  doi: 10.1007/s11856-015-1251-z.  Google Scholar

[13]

J. García-Melián and J. Sabina de Lis, Stationary profiles of degenerate problems when a parameter is large, Differential Integral Equations, 13 (2000), 1201-1232.   Google Scholar

[14]

F. GazzolaH.-C. Grunau and M. Squassina, Existence and nonexistence results for critical growth biharmonic elliptic equations, Calc. Var. Partial Differential Equations, 18 (2003), 117-143.  doi: 10.1007/s00526-002-0182-9.  Google Scholar

[15]

F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, vol. 1991 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010, Positivity preserving and nonlinear higher order elliptic equations in bounded domains. doi: 10.1007/978-3-642-12245-3.  Google Scholar

[16]

M. Guedda and L. Véron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal., 13 (1989), 879-902.  doi: 10.1016/0362-546X(89)90020-5.  Google Scholar

[17]

P. Hess, On multiple positive solutions of nonlinear elliptic eigenvalue problems, Comm. Partial Differential Equations, 6 (1981), 951-961.  doi: 10.1080/03605308108820200.  Google Scholar

[18]

J. Hulshof and R. van der Vorst, Differential systems with strongly indefinite variational structure, J. Funct. Anal., 114 (1993), 32-58.  doi: 10.1006/jfan.1993.1062.  Google Scholar

[19]

L. IturriagaS. Lorca and E. Massa, Positive solutions for the p-Laplacian involving critical and supercritical nonlinearities with zeros, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 763-771.  doi: 10.1016/j.anihpc.2009.11.003.  Google Scholar

[20]

L. IturriagaS. Lorca and E. Massa, Multiple positive solutions for the m-Laplacian and a nonlinearity with many zeros, Differential Integral Equations, 30 (2017), 145-159.   Google Scholar

[21]

L. IturriagaE. MassaJ. Sánchez and P. Ubilla, Positive solutions of the p-Laplacian involving a superlinear nonlinearity with zeros, J. Differential Equations, 248 (2010), 309-327.  doi: 10.1016/j.jde.2009.08.008.  Google Scholar

[22]

L. IturriagaE. MassaJ. Sanchez and P. Ubilla, Positive solutions for an elliptic equation in an annulus with a superlinear nonlinearity with zeros, Math. Nach., 287 (2014), 1131-1141.  doi: 10.1002/mana.201100285.  Google Scholar

[23]

A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev., 32 (1990), 537-578.  doi: 10.1137/1032120.  Google Scholar

[24]

A. C. Lazer and P. J. McKenna, Global bifurcation and a theorem of Tarantello, J. Math. Anal. Appl., 181 (1994), 648-655.  doi: 10.1006/jmaa.1994.1049.  Google Scholar

[25]

P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467.  doi: 10.1137/1024101.  Google Scholar

[26]

Z. Liu, Positive solutions of superlinear elliptic equations, J. Funct. Anal., 167 (1999), 370-398.  doi: 10.1006/jfan.1999.3446.  Google Scholar

[27]

Z. Liu, Positive solutions of a class of nonlinear elliptic eigenvalue problems, Math. Z., 242 (2002), 663-686.  doi: 10.1007/s002090100373.  Google Scholar

[28]

A. M. Micheletti and A. Pistoia, Multiplicity results for a fourth-order semilinear elliptic problem, Nonlinear Anal., 31 (1998), 895-908.  doi: 10.1016/S0362-546X(97)00446-X.  Google Scholar

[29]

A. M. Micheletti and A. Pistoia, Nontrivial solutions for some fourth order semilinear elliptic problems, Nonlinear Anal., 34 (1998), 509-523.  doi: 10.1016/S0362-546X(97)00596-8.  Google Scholar

[30]

E. S. NoussairC. A. Swanson and J. Yang, Critical semilinear biharmonic equations in RN, Proc. Roy. Soc. Edinburgh Sect. A, 121 (1992), 139-148.  doi: 10.1017/S0308210500014189.  Google Scholar

[31]

L. A. Peletier and R. van der Vorst, Existence and nonexistence of positive solutions of nonlinear elliptic systems and the biharmonic equation, Differential Integral Equations, 5 (1992), 747-767.   Google Scholar

[32]

P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl. (9), 69 (1990), 55-83.   Google Scholar

[33]

S. Takeuchi, Coincidence sets in semilinear elliptic problems of logistic type, Differential Integral Equations, 20 (2007), 1075-1080.   Google Scholar

[34]

S. Takeuchi, Partial flat core properties associated to the p-Laplace operator, Discrete Contin. Dyn. Syst., (2007), 965-973.   Google Scholar

[35]

G. Tarantello, A note on a semilinear elliptic problem, Differential Integral Equations, 5 (1992), 561-565.   Google Scholar

[36]

G. Xu and J. Zhang, Existence results for some fourth-order nonlinear elliptic problems of local superlinearity and sublinearity, J. Math. Anal. Appl., 281 (2003), 633-640.  doi: 10.1016/S0022-247X(03)00170-7.  Google Scholar

[37]

J. Zhang and Z. Wei, Multiple solutions for a class of biharmonic equations with a nonlinearity concave at the origin, J. Math. Anal. Appl., 383 (2011), 291-306.  doi: 10.1016/j.jmaa.2011.05.030.  Google Scholar

[38]

Y. Zhang, Positive solutions of semilinear biharmonic equations with critical Sobolev exponents, Nonlinear Anal., 75 (2012), 55-67.  doi: 10.1016/j.na.2011.07.065.  Google Scholar

[1]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[2]

Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036

[3]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[4]

Shengbing Deng, Tingxi Hu, Chun-Lei Tang. $ N- $Laplacian problems with critical double exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 987-1003. doi: 10.3934/dcds.2020306

[5]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[6]

Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020294

[7]

Philippe Laurençot, Christoph Walker. Variational solutions to an evolution model for MEMS with heterogeneous dielectric properties. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 677-694. doi: 10.3934/dcdss.2020360

[8]

Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439

[9]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[10]

Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325

[11]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[12]

Tong Yang, Seiji Ukai, Huijiang Zhao. Stationary solutions to the exterior problems for the Boltzmann equation, I. Existence. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 495-520. doi: 10.3934/dcds.2009.23.495

[13]

Ryuji Kajikiya. Existence of nodal solutions for the sublinear Moore-Nehari differential equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1483-1506. doi: 10.3934/dcds.2020326

[14]

Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1723-1747. doi: 10.3934/dcdsb.2020180

[15]

Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020293

[16]

Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020403

[17]

Yutong Chen, Jiabao Su. Nontrivial solutions for the fractional Laplacian problems without asymptotic limits near both infinity and zero. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021007

[18]

Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265

[19]

Dorothee Knees, Chiara Zanini. Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 121-149. doi: 10.3934/dcdss.2020332

[20]

Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (161)
  • HTML views (156)
  • Cited by (0)

Other articles
by authors

[Back to Top]