×

Existence of groundstates for a class of nonlinear Choquard equations. (English) Zbl 1325.35052

Summary: We prove the existence of a nontrivial solution \( u \in H^1 (\mathbb{R}^N)\) to the nonlinear Choquard equation \[ - \Delta u + u = \bigl (I_\alpha \ast F (u)\bigr ) F'(u)\quad \text{in }\mathbb R^N, \] where \( I_\alpha \) is a Riesz potential, under almost necessary conditions on the nonlinearity \( F\) in the spirit of Berestycki and Lions. This solution is a groundstate and has additional local regularity properties; if moreover \( F\) is even and monotone on \( (0,\infty )\), then \( u\) is of constant sign and radially symmetric.

MSC:

35J61 Semilinear elliptic equations
35B33 Critical exponents in context of PDEs
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
35B65 Smoothness and regularity of solutions to PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
45K05 Integro-partial differential equations
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Azzollini, A.; d’Avenia, P.; Pomponio, A., Multiple critical points for a class of nonlinear functionals, Ann. Mat. Pura Appl. (4), 190, 3, 507-523 (2011) · Zbl 1230.58014
[2] Azzollini, A.; d’Avenia, P.; Pomponio, A., On the Schr\`“odinger-Maxwell equations under the effect of a general nonlinear term, Ann. Inst. H. Poincar\'”e Anal. Non Lin\'eaire, 27, 2, 779-791 (2010) · Zbl 1187.35231
[3] Baernstein, Albert, II, A unified approach to symmetrization. Partial differential equations of elliptic type, Cortona, 1992, Sympos. Math., XXXV, 47-91 (1994), Cambridge Univ. Press, Cambridge · Zbl 0830.35005
[4] Bartsch, Thomas; de Valeriola, S{\'e}bastien, Normalized solutions of nonlinear Schr\"odinger equations, Arch. Math. (Basel), 100, 1, 75-83 (2013) · Zbl 1260.35098
[5] Bartsch, Thomas; Weth, Tobias; Willem, Michel, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math., 96, 1-18 (2005) · Zbl 1206.35086
[6] Berestycki, H.; Lions, P.-L., Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82, 4, 313-345 (1983) · Zbl 0533.35029
[7] Brezis, Haim, Functional analysis, Sobolev spaces and partial differential equations, Universitext, xiv+599 pp. (2011), Springer, New York · Zbl 1220.46002
[8] Br{\'e}zis, Ha{\`“{\i }}m; Kato, Tosio, Remarks on the Schr\'”odinger operator with singular complex potentials, J. Math. Pures Appl. (9), 58, 2, 137-151 (1979) · Zbl 0408.35025
[9] Brock, Friedemann; Solynin, Alexander Yu., An approach to symmetrization via polarization, Trans. Amer. Math. Soc., 352, 4, 1759-1796 (2000) · Zbl 0965.49001
[10] Byeon, Jaeyoung; Jeanjean, Louis; Mari{\c{s}}, Mihai, Symmetry and monotonicity of least energy solutions, Calc. Var. Partial Differential Equations, 36, 4, 481-492 (2009) · Zbl 1226.35041
[11] Choquard, Philippe; Stubbe, Joachim; Vuffray, Marc, Stationary solutions of the Schr\"odinger-Newton model-an ODE approach, Differential Integral Equations, 21, 7-8, 665-679 (2008) · Zbl 1224.35385
[12] Cingolani, Silvia; Clapp, M{\'o}nica; Secchi, Simone, Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63, 2, 233-248 (2012) · Zbl 1247.35141
[13] Cingolani, Silvia; Secchi, Simone; Squassina, Marco, Semi-classical limit for Schr\"odinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 140, 5, 973-1009 (2010) · Zbl 1215.35146
[14] Flucher, M.; M{\"u}ller, S., Radial symmetry and decay rate of variational ground states in the zero mass case, SIAM J. Math. Anal., 29, 3, 712-719 (electronic) (1998) · Zbl 0908.35005
[15] Genev, Hristo; Venkov, George, Soliton and blow-up solutions to the time-dependent Schr\"odinger-Hartree equation, Discrete Contin. Dyn. Syst. Ser. S, 5, 5, 903-923 (2012) · Zbl 1247.35143
[16] Gidas, B.; Ni, Wei Ming; Nirenberg, L., Symmetry of positive solutions of nonlinear elliptic equations in \({\bf R}^n\). Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud. 7, 369-402 (1981), Academic Press, New York-London
[17] Hirata, Jun; Ikoma, Norihisa; Tanaka, Kazunaga, Nonlinear scalar field equations in \(\mathbb{R}^N\): mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal., 35, 2, 253-276 (2010) · Zbl 1203.35106
[18] Jeanjean, Louis, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28, 10, 1633-1659 (1997) · Zbl 0877.35091
[19] Jeanjean, Louis, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on \({\bf R}^N\), Proc. Roy. Soc. Edinburgh Sect. A, 129, 4, 787-809 (1999) · Zbl 0935.35044
[20] Jeanjean, Louis; Tanaka, Kazunaga, A remark on least energy solutions in \({\bf R}^N\), Proc. Amer. Math. Soc., 131, 8, 2399-2408 (electronic) (2003) · Zbl 1094.35049
[21] Kavian, Otared, Introduction \`“a la th\'”eorie des points critiques et applications aux probl\`“emes elliptiques, Math\'”ematiques & Applications (Berlin) [Mathematics & Applications] 13, viii+325 pp. (1993), Springer-Verlag, Paris · Zbl 0797.58005
[22] Lieb, Elliott H., Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Studies in Appl. Math., 57, 2, 93-105 (1976/77) · Zbl 0369.35022
[23] Lieb, Elliott H.; Loss, Michael, Analysis, Graduate Studies in Mathematics 14, xxii+346 pp. (2001), American Mathematical Society, Providence, RI · Zbl 0966.26002
[24] Lions, P.-L., The Choquard equation and related questions, Nonlinear Anal., 4, 6, 1063-1072 (1980) · Zbl 0453.47042
[25] Lions, Pierre-Louis, Sym\'etrie et compacit\'e dans les espaces de Sobolev, J. Funct. Anal., 49, 3, 315-334 (1982) · Zbl 0501.46032
[26] Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 1, 4, 223-283 (1984) · Zbl 0704.49004
[27] Ma, Li; Zhao, Lin, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195, 2, 455-467 (2010) · Zbl 1185.35260
[28] Menzala, Gustavo Perla, On regular solutions of a nonlinear equation of Choquard’s type, Proc. Roy. Soc. Edinburgh Sect. A, 86, 3-4, 291-301 (1980) · Zbl 0449.35034
[29] Menzala, Gustavo Perla, On the nonexistence of solutions for an elliptic problem in unbounded domains, Funkcial. Ekvac., 26, 3, 231-235 (1983) · Zbl 0557.35046
[30] Moroz, Irene M.; Penrose, Roger; Tod, Paul, Spherically-symmetric solutions of the Schr\"odinger-Newton equations, Classical Quantum Gravity, 15, 9, 2733-2742 (1998) · Zbl 0936.83037
[31] Moroz, Vitaly; Van Schaftingen, Jean, Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265, 2, 153-184 (2013) · Zbl 1285.35048
[32] Pekar, S., Untersuchung {\"u}ber die Elektronentheorie der Kristalle, 184 pp. (1954), Akademie Verlag: Berlin:Akademie Verlag · Zbl 0058.45503
[33] P{\'o}lya, G.; Szeg{\"o}, G., Inequalities for the capacity of a condenser, Amer. J. Math., 67, 1-32 (1945) · Zbl 0063.06304
[34] P{\'o}lya, G.; Szeg{\"o}, G., Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27, xvi+279 pp. (1951), Princeton University Press, Princeton, N. J.
[35] Squassina, Marco; Van Schaftingen, Jean, Finding critical points whose polarization is also a critical point, Topol. Methods Nonlinear Anal., 40, 2, 371-379 (2012) · Zbl 1283.35023
[36] Strauss, Walter A., Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55, 2, 149-162 (1977) · Zbl 0356.35028
[37] Struwe, Michael, Variational methods, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 34, xx+302 pp. (2008), Springer-Verlag, Berlin · Zbl 1284.49004
[38] Tod, Paul; Moroz, Irene M., An analytical approach to the Schr\"odinger-Newton equations, Nonlinearity, 12, 2, 201-216 (1999) · Zbl 0942.35077
[39] Trudinger, Neil S., Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa (3), 22, 265-274 (1968) · Zbl 0159.23801
[40] Van Schaftingen, Jean, Symmetrization and minimax principles, Commun. Contemp. Math., 7, 4, 463-481 (2005) · Zbl 1206.35088
[41] Van Schaftingen, J.; Willem, M., Set transformations, symmetrizations and isoperimetric inequalities. Nonlinear analysis and applications to physical sciences, 135-152 (2004), Springer Italia, Milan · Zbl 1453.26034
[42] Van Schaftingen, Jean; Willem, Michel, Symmetry of solutions of semilinear elliptic problems, J. Eur. Math. Soc. (JEMS), 10, 2, 439-456 (2008) · Zbl 1148.35025
[43] Willem, Michel, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, 24, x+162 pp. (1996), Birkh\"auser Boston, Inc., Boston, MA · Zbl 0856.49001
[44] Willem, Michel, Functional analysis: Fundamentals and applications, Cornerstones, xiv+213 pp. (2013), Birkh\"auser/Springer, New York · Zbl 1284.46001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.