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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 & 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
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