American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Schauder estimates for solutions of linear parabolic integro-differential equations
  • DCDS Home
  • This Issue
  • Next Article
    On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density
December  2015, 35(12): 5963-5976. doi: 10.3934/dcds.2015.35.5963

Ground states for scalar field equations with anisotropic nonlocal nonlinearities

Antonio Iannizzotto 1, , Kanishka Perera 2, and  Marco Squassina 3,

1. 

Department of Mathematics and Computer Science, University of Cagliari, Viale L. Merello 92, 09123 Cagliari, Italy
2. 

Department of Mathematical Sciences, Florida Institute of Technology, 150 W University Blvd, Melbourne, FL 32901
3. 

Dipartimento di Informatica, Università di Verona, Strada Le Grazie 15, 37134 Verona

Received  January 2014 Published  May 2015

We consider a class of scalar field equations with anisotropic nonlocal nonlinearities. We obtain a suitable extension of the well-known compactness lemma of Benci and Cerami to this variable exponent setting, and use it to prove that the Palais-Smale condition holds at all level below a certain threshold. We deduce the existence of a ground state when the variable exponent slowly approaches the limit at infinity from below.
Keywords: anisotropic nonlocal nonlinearity, variable exponent, loss of compactness, Scalar field equation, existence of ground state..
Mathematics Subject Classification: Primary: 35J20, 46B50; Secondary: 74G6.
Citation: Antonio Iannizzotto, Kanishka Perera, Marco Squassina. Ground states for scalar field equations with anisotropic nonlocal nonlinearities. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5963-5976. doi: 10.3934/dcds.2015.35.5963
References:
[1]

A. Bahri and P.-L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 365.  doi: 10.1016/S0294-1449(97)80142-4.  Google Scholar

[2]

V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains,, Arch. Rational Mech. Anal., 99 (1987), 283.  doi: 10.1007/BF00282048.  Google Scholar

[3]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313.  doi: 10.1007/BF00250555.  Google Scholar

[4]

J. Byeon, L. Jeanjean and M. Mariş, Symmetry and monotonicity of least energy solutions,, Calc. Var. Partial Differential Equations, 36 (2009), 481.  doi: 10.1007/s00526-009-0238-1.  Google Scholar

[5]

L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents,, Lecture Notes in Mathematics, (2017).  doi: 10.1007/978-3-642-18363-8.  Google Scholar

[6]

X. Fan and D. Zhao, On the spaces $L^{p(\cdot)}(\Omega)$ and $W^{m,p(x)}(\Omega)$,, J. Math. Anal. Appl., 263 (2001), 424.  doi: 10.1006/jmaa.2000.7617.  Google Scholar

[7]

G. Franzina and P. Lindqvist, An eigenvalue problem with variable exponents,, Nonlinear Anal., 85 (2013), 1.  doi: 10.1016/j.na.2013.02.011.  Google Scholar

[8]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbbR^n$,, Arch. Rational Mech. Anal., 105 (1989), 243.  doi: 10.1007/BF00251502.  Google Scholar

[9]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case I,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109.   Google Scholar

[10]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case II,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223.   Google Scholar

[11]

P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems,, Comm. Math. Phys., 109 (1987), 33.  doi: 10.1007/BF01205672.  Google Scholar

[12]

M. Mariş, On the symmetry of minimizers,, Arch. Rational Mech. Anal., 192 (2009), 311.  doi: 10.1007/s00205-008-0136-2.  Google Scholar

[13]

S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 12 (1995), 319.   Google Scholar

[14]

M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,, Ergebnisse der Mathematik und ihrer Grenzgebiete, (2008).   Google Scholar

[15]

K. Tintarev and K.-H. Fieseler, Concentration Compactness. Functional-Analytic Grounds and Applications,, Imperial College Press, (2007).   Google Scholar

show all references

References:
[1]

A. Bahri and P.-L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 365.  doi: 10.1016/S0294-1449(97)80142-4.  Google Scholar

[2]

V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains,, Arch. Rational Mech. Anal., 99 (1987), 283.  doi: 10.1007/BF00282048.  Google Scholar

[3]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313.  doi: 10.1007/BF00250555.  Google Scholar

[4]

J. Byeon, L. Jeanjean and M. Mariş, Symmetry and monotonicity of least energy solutions,, Calc. Var. Partial Differential Equations, 36 (2009), 481.  doi: 10.1007/s00526-009-0238-1.  Google Scholar

[5]

L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents,, Lecture Notes in Mathematics, (2017).  doi: 10.1007/978-3-642-18363-8.  Google Scholar

[6]

X. Fan and D. Zhao, On the spaces $L^{p(\cdot)}(\Omega)$ and $W^{m,p(x)}(\Omega)$,, J. Math. Anal. Appl., 263 (2001), 424.  doi: 10.1006/jmaa.2000.7617.  Google Scholar

[7]

G. Franzina and P. Lindqvist, An eigenvalue problem with variable exponents,, Nonlinear Anal., 85 (2013), 1.  doi: 10.1016/j.na.2013.02.011.  Google Scholar

[8]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbbR^n$,, Arch. Rational Mech. Anal., 105 (1989), 243.  doi: 10.1007/BF00251502.  Google Scholar

[9]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case I,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109.   Google Scholar

[10]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case II,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223.   Google Scholar

[11]

P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems,, Comm. Math. Phys., 109 (1987), 33.  doi: 10.1007/BF01205672.  Google Scholar

[12]

M. Mariş, On the symmetry of minimizers,, Arch. Rational Mech. Anal., 192 (2009), 311.  doi: 10.1007/s00205-008-0136-2.  Google Scholar

[13]

S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 12 (1995), 319.   Google Scholar

[14]

M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,, Ergebnisse der Mathematik und ihrer Grenzgebiete, (2008).   Google Scholar

[15]

K. Tintarev and K.-H. Fieseler, Concentration Compactness. Functional-Analytic Grounds and Applications,, Imperial College Press, (2007).   Google Scholar

[1]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309
[2]

Christophe Zhang. Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021006
[3]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109
[4]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[5]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912
[6]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1757-1778. doi: 10.3934/dcdss.2020453
[7]

Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021012
[8]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298
[9]

Todd Hurst, Volker Rehbock. Optimizing micro-algae production in a raceway pond with variable depth. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021027
[10]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450
[11]

Qian Liu. The lower bounds on the second-order nonlinearity of three classes of Boolean functions. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020136
[12]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089
[13]

Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021006
[14]

Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021011
[15]

Jia Cai, Guanglong Xu, Zhensheng Hu. Sketch-based image retrieval via CAT loss with elastic net regularization. Mathematical Foundations of Computing, 2020, 3 (4) : 219-227. doi: 10.3934/mfc.2020013
[16]

Shanshan Chen, Junping Shi, Guohong Zhang. Spatial pattern formation in activator-inhibitor models with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1843-1866. doi: 10.3934/dcdsb.2020042
[17]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035
[18]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810
[19]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973
[20]

Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (24)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences