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December  2015, 35(12): 5963-5976. doi: 10.3934/dcds.2015.35.5963

Ground states for scalar field equations with anisotropic nonlocal nonlinearities

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.
Citation: Antonio Iannizzotto, Kanishka Perera, Marco Squassina. Ground states for scalar field equations with anisotropic nonlocal nonlinearities. Discrete and Continuous Dynamical Systems, 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), 365-413. doi: 10.1016/S0294-1449(97)80142-4.

[2]

V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Anal., 99 (1987), 283-300. doi: 10.1007/BF00282048.

[3]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.

[4]

J. Byeon, L. Jeanjean and M. Mariş, Symmetry and monotonicity of least energy solutions, Calc. Var. Partial Differential Equations, 36 (2009), 481-492. doi: 10.1007/s00526-009-0238-1.

[5]

L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, 2017, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.

[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), 424-446. doi: 10.1006/jmaa.2000.7617.

[7]

G. Franzina and P. Lindqvist, An eigenvalue problem with variable exponents, Nonlinear Anal., 85 (2013), 1-16. doi: 10.1016/j.na.2013.02.011.

[8]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.

[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), 109-145.

[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), 223-283.

[11]

P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33-97. doi: 10.1007/BF01205672.

[12]

M. Mariş, On the symmetry of minimizers, Arch. Rational Mech. Anal., 192 (2009), 311-330. doi: 10.1007/s00205-008-0136-2.

[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), 319-337.

[14]

M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin, 2008.

[15]

K. Tintarev and K.-H. Fieseler, Concentration Compactness. Functional-Analytic Grounds and Applications, Imperial College Press, London, 2007.

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), 365-413. doi: 10.1016/S0294-1449(97)80142-4.

[2]

V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Anal., 99 (1987), 283-300. doi: 10.1007/BF00282048.

[3]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.

[4]

J. Byeon, L. Jeanjean and M. Mariş, Symmetry and monotonicity of least energy solutions, Calc. Var. Partial Differential Equations, 36 (2009), 481-492. doi: 10.1007/s00526-009-0238-1.

[5]

L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, 2017, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.

[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), 424-446. doi: 10.1006/jmaa.2000.7617.

[7]

G. Franzina and P. Lindqvist, An eigenvalue problem with variable exponents, Nonlinear Anal., 85 (2013), 1-16. doi: 10.1016/j.na.2013.02.011.

[8]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.

[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), 109-145.

[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), 223-283.

[11]

P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33-97. doi: 10.1007/BF01205672.

[12]

M. Mariş, On the symmetry of minimizers, Arch. Rational Mech. Anal., 192 (2009), 311-330. doi: 10.1007/s00205-008-0136-2.

[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), 319-337.

[14]

M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin, 2008.

[15]

K. Tintarev and K.-H. Fieseler, Concentration Compactness. Functional-Analytic Grounds and Applications, Imperial College Press, London, 2007.

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