# American Institute of Mathematical Sciences

• 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

 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
 [1] Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253 [2] Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243 [3] Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116 [4] Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270 [5] Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259 [6] Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276 [7] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264 [8] Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453 [9] Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020385 [10] Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164 [11] 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 [12] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450 [13] Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018 [14] Illés Horváth, Kristóf Attila Horváth, Péter Kovács, Miklós Telek. Mean-field analysis of a scaling MAC radio protocol. Journal of Industrial & Management Optimization, 2021, 17 (1) : 279-297. doi: 10.3934/jimo.2019111 [15] Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103 [16] Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260 [17] Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110 [18] Guangbin CAI, Yang Zhao, Wanzhen Quan, Xiusheng Zhang. Design of LPV fault-tolerant controller for hypersonic vehicle based on state observer. Journal of Industrial & Management Optimization, 2021, 17 (1) : 447-465. doi: 10.3934/jimo.2019120 [19] Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316 [20] Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462

2019 Impact Factor: 1.338