## Least action nodal solutions for the quadratic Choquard equation.(English)Zbl 1355.35079

Summary: We prove the existence of a minimal action nodal solution for the quadratic Choquard equation $-\Delta u + u = \bigl (I_\alpha \ast \| u\|^2\bigr )u \quad \text{in } \mathbb{R}^N,$ where $$I_{\alpha}$$ is the Riesz potential of order $$\alpha \in (0,N)$$. The solution is constructed as the limit of minimal action nodal solutions for the nonlinear Choquard equations $-\Delta u + u = \bigl (I_{\alpha} \ast \| u\|^p\bigr )| u|^{p-2}u \quad \text{in } \mathbb{R}^N$ when $$p\searrow 2$$. The existence of minimal action nodal solutions for $$p>2$$ can be proved using a variational minimax procedure over a Nehari nodal set. No minimal action nodal solutions exist when $$p<2$$.

### MSC:

 35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian 35J20 Variational methods for second-order elliptic equations 35Q55 NLS equations (nonlinear Schrödinger equations)
Full Text:

### References:

 [1] Ackermann, Nils; Weth, Tobias, Multibump solutions of nonlinear periodic Schr\"odinger equations in a degenerate setting, Commun. Contemp. Math., 7, 3, 269-298 (2005) · Zbl 1070.35083 [2] Bogachev, V. I., Measure theory. Vol. I, II, Vol. I: xviii+500 pp., Vol. II: xiv+575 pp. (2007), Springer-Verlag, Berlin [3] Bonanno, Claudio; d’Avenia, Pietro; Ghimenti, Marco; Squassina, Marco, Soliton dynamics for the generalized Choquard equation, J. Math. Anal. Appl., 417, 1, 180-199 (2014) · Zbl 1332.35066 [4] Castro, Alfonso; Cossio, Jorge; Neuberger, John M., A sign-changing solution for a superlinear Dirichlet problem, Rocky Mountain J. Math., 27, 4, 1041-1053 (1997) · Zbl 0907.35050 [5] Castro, Alfonso; Cossio, Jorge; Neuberger, John M., A minmax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems, Electron. J. Differential Equations, No. 02, 18 pp. (electronic) pp. (1998) · Zbl 0901.35028 [6] Cerami, G.; Solimini, S.; Struwe, M., Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal., 69, 3, 289-306 (1986) · Zbl 0614.35035 [7] 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 [8] Cingolani, Silvia; Secchi, Simone, Multiple $$\mathbb{S}^1$$-orbits for the Schr\"odinger-Newton system, Differential Integral Equations, 26, 9-10, 867-884 (2013) · Zbl 1299.35281 [9] Cingolani, Silvia; Secchi, Simone, Ground states for the pseudo-relativistic Hartree equation with external potential, Proc. Roy. Soc. Edinburgh Sect. A, 145, 1, 73-90 (2015) · Zbl 1320.35300 [10] Clapp, M{\'o}nica; Salazar, Dora, Positive and sign changing solutions to a nonlinear Choquard equation, J. Math. Anal. Appl., 407, 1, 1-15 (2013) · Zbl 1310.35114 [11] d’Avenia, Pietro; Siciliano, Gaetano; Squassina, Marco, On fractional Choquard equations, Math. Models Methods Appl. Sci., 25, 8, 1447-1476 (2015) · Zbl 1323.35205 [12] Devreese, Jozef T.; Alexandrov, Alexandre S., Advances in polaron physics, Springer Series in Solid-State Sciences 159, ix+167 pp. (2010), Springer [13] Fr{\`“o}hlich, J{\'”u}rg; Lenzmann, Enno, Mean-field limit of quantum Bose gases and nonlinear Hartree equation. S\'eminaire: \'Equations aux D\'eriv\'ees Partielles. 2003-2004, S\'emin. \'Equ. D\'eriv. Partielles, Exp. No. XIX, 26 pp. (2004), \'Ecole Polytech., Palaiseau · Zbl 1292.81040 [14] Ghimenti, Marco; Van Schaftingen, Jean, Nodal solutions for the Choquard equation, J. Funct. Anal., 271, 1, 107-135 (2016) · Zbl 1345.35046 [15] Jones, K. R. W., Newtonian Quantum Gravity, Australian Journal of Physics, 48, 6, 1055-1081 (1995) [16] Lenzmann, Enno, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2, 1, 1-27 (2009) · Zbl 1183.35266 [17] 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 [18] Lieb, Elliott H.; Loss, Michael, Analysis, Graduate Studies in Mathematics 14, xxii+346 pp. (2001), American Mathematical Society, Providence, RI · Zbl 0966.26002 [19] Lions, P.-L., The Choquard equation and related questions, Nonlinear Anal., 4, 6, 1063-1072 (1980) · Zbl 0453.47042 [20] Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 1, 2, 109-145 (1984) · Zbl 0541.49009 [21] Lions, Pierre-Louis, 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 [22] Liu, Zhaoli; Wang, Zhi-Qiang, On the Ambrosetti-Rabinowitz superlinear condition, Adv. Nonlinear Stud., 4, 4, 563-574 (2004) · Zbl 1113.35048 [23] 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 [24] 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 [25] 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 [26] 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 [27] Pekar, S., Untersuchung {\"u}ber die Elektronentheorie der Kristalle, 184 pp. (1954), Akademie Verlag: Berlin:Akademie Verlag · Zbl 0058.45503 [28] Van Schaftingen, Jean, Interpolation inequalities between Sobolev and Morrey-Campanato spaces: a common gateway to concentration-compactness and Gagliardo-Nirenberg interpolation inequalities, Port. Math., 71, 3-4, 159-175 (2014) · Zbl 1326.46032 [29] 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 [30] Weth, Tobias, Spectral and variational characterizations of solutions to semilinear eigenvalue problems (2001), Mainz [31] Weth, Tobias, Energy bounds for entire nodal solutions of autonomous superlinear equations, Calc. Var. Partial Differential Equations, 27, 4, 421-437 (2006) · Zbl 1151.35365 [32] 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 [33] Willem, Michel, Functional analysis, 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.