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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 $\mathbbR^n$, 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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##### 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 $\mathbbR^n$, 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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