Ground states for scalar field equations with anisotropic nonlocal nonlinearities

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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

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

Abstract
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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 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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