×

Ground state solutions for a class of strongly indefinite Choquard equations. (English) Zbl 1440.35126

Summary: In this paper, we study the existence and concentration of ground state solution for the Choquard equation
\[\begin{cases}-\Delta u+V(x)u=\left(\int_{\mathbb{R}^N}\frac{A(\epsilon y)|u(y)|^p}{|x-y|^\mu}\text{d}y\right)A(\epsilon x)|u|^{p-2}u\quad\text{in }\mathbb{R}^N,\\u\in H^1(\mathbb{R}^N),\end{cases}\]
where \(N\ge 2\), \(0<\mu <2\), \(\epsilon\) is a positive parameter. \(V\) is a \(\mathbb{Z}^N\)-periodic function, and 0 lies in a gap of the spectrum of \(-\Delta+V\). \(A\in C(\mathbb{R}^N)\) satisfies
\(\begin{aligned}0<\inf\limits_{x\in\mathbb{R}^N}A(x)\le\lim\limits_{|x|\rightarrow+\infty}A(x) <\sup\limits_{x\in\mathbb{R}^N}A(x).\end{aligned}\)

MSC:

35J61 Semilinear elliptic equations
35J50 Variational methods for elliptic systems
58E30 Variational principles in infinite-dimensional spaces
35A01 Existence problems for PDEs: global existence, local existence, non-existence
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ackermann, N., A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal., 234, 423-443 (2006) · Zbl 1126.35057
[2] Alves, C.O., Germano, G.F.: Existence and concentration of ground state solution for a class of indefinite variational problem. arXiv preprint arXiv:1801.06872 (2018) · Zbl 1394.35151
[3] Alves, C.O., Germano, G.F.: Existence and concentration phenomena for a class of indefinite variational problems with critical growth. arXiv preprint arXiv:1801.08138 (2018) · Zbl 1433.35022
[4] Alves, CO; Yang, M., Existence of semiclassical ground state solutions for a generalized Choquard equation, J. Differ. Equ., 257, 4133-4164 (2014) · Zbl 1309.35036
[5] Alves, CO; Cassani, D.; Tarsi, C.; Yang, M., Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in \({\mathbb{R}}^2 \), J. Differ. Equ., 261, 1933-1972 (2016) · Zbl 1347.35096
[6] Ambrosetti, A.; Badiale, M.; Cingolani, S., Semiclassical states of nonlinear Schödinger equations, Arch. Ration. Mech. Anal., 140, 285-300 (1997) · Zbl 0896.35042
[7] Buffoni, B.; Jeanjean, L.; Stuart, CA, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Am. Math. Soc., 119, 1, 179-186 (1993) · Zbl 0789.35052
[8] Choquard, P.; Stubbe, J.; Vuffray, M., Stationary solutions of the Schrödinger-Newton model-an ODE approach, Differ. Integral Equ., 21, 665-679 (2008) · Zbl 1224.35385
[9] del Pino, M.; Felmer, P., Multipeak bound states of nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15, 127-149 (1998) · Zbl 0901.35023
[10] del Pino, M.; Felmer, P., Local Mountain Pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4, 121-137 (1996) · Zbl 0844.35032
[11] Ding, YH, Semi-classical ground states concentrating on the nonlinear potential for a Dirac equation, J. Differ. Equ., 249, 1015-1034 (2010) · Zbl 1193.35161
[12] Ding, YH; Wei, JC, Stationary states of nonlinear Dirac equations with general potentials, Rev. Math. Phys., 20, 1007-1032 (2008) · Zbl 1170.35082
[13] Ding, YH; Lee, C.; Zhao, F., Semiclassical limits of ground state solutions to Schrödinger systems, Calc. Var. Partial Differ. Equ., 51, 725-760 (2014) · Zbl 1310.35104
[14] Floer, A.; Weinstein, A., Nonspreading wave pachets for the packets for the cubic Schrödinger with a bounded potential, J. Funct. Anal., 69, 397-408 (1986) · Zbl 0613.35076
[15] Gao, F., Yang, M.: On the Brézis-Nirenberg type critical problem for nonlinear Choquard equation. Sci. China Math. 10.1007/s11425-016-9067-5 · Zbl 1397.35087
[16] Gao, F.; Yang, M., A strongly indefinite Choquard equation with critical exponent due to the Hardy-Littlewood-Sobolev inequality, Commun. Contemp. Math., 20, 1750037 (2018) · Zbl 1391.35126
[17] Gui, C.; Wei, J., Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differ. Equ., 158, 1-27 (1999) · Zbl 1061.35502
[18] Jeanjean, L.; Tanaka, K., Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. Var. Partial Differ. Equ., 21, 287-318 (2004) · Zbl 1060.35012
[19] Kryszewski, W.; Szulkin, A., Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differ. Equ., 3, 441-472 (1998) · Zbl 0947.35061
[20] Li, G.; Szulkin, A., An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4, 763-776 (2002) · Zbl 1056.35065
[21] Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57 93-105 (1976/77) · Zbl 0369.35022
[22] Lions, P-L, The Choquard equation and related questions, Nonlinear Anal., 4, 1063-1072 (1980) · Zbl 0453.47042
[23] Liu, M.; Tang, Z., Multiplicity and concentration of solutions for Choquard equation via Nehari method and pseudo-index theory, Discrete Contin. Dyn. Syst., 39, 6, 3365-3398 (2019) · Zbl 1415.35108
[24] Liu, M., Tang, Z.: Pseudo-index theory and Nehari method for a fractional Choquard equation. Pac. J. Math. (to appear) · Zbl 1437.35304
[25] Luo, H., Ground state solutions of Pohoz̆aev type and Nehari type for a class of nonlinear Choquard equations, J. Math. Anal. Appl., 467, 2, 842-862 (2018) · Zbl 1398.35071
[26] Ma, L.; Zhao, L., Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195, 455-467 (2010) · Zbl 1185.35260
[27] Menzala, GP, On regular solutions of a nonlinear equation of Choquard’s type, Proc. R. Soc. Edinb. Sect. A, 86, 291-301 (1980) · Zbl 0449.35034
[28] Moroz, V.; Van Schaftingen, J., Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265, 153-184 (2013) · Zbl 1285.35048
[29] Moroz, IM; Penrose, R.; Tod, P., Spherically-symmetric solutions of the Schrödinger-Newton equations, Class. Quantum Gravity, 15, 2733-2742 (1998) · Zbl 0936.83037
[30] Pankov, A., Periodic nonlinear Schrödinger equations with application to photonic crystals, Milan J. Math., 73, 259-287 (2005) · Zbl 1225.35222
[31] Pankov, A.; Pflüger, K., On a semilinear Schrödinger equation with periodic potential, Nonlinear Anal. TMA, 33, 593-609 (1998) · Zbl 0952.35047
[32] Pekar, S., Untersuchung über die Elektronentheorie der Kristalle (1954), Berlin: Akademie Verlag, Berlin · Zbl 0058.45503
[33] Rabinowitz, P., On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43, 270-291 (1992) · Zbl 0763.35087
[34] Riesz, M., L’intégrale de Riemann-Liouville et le problème de Cauchy, Acta Math., 81, 1-223 (1949) · Zbl 0033.27601
[35] Szulkin, A.; Weth, T., Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257, 3802-3822 (2009) · Zbl 1178.35352
[36] Tang, XH, Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58, 715-728 (2015) · Zbl 1321.35055
[37] Tod, P.; Moroz, IM, An analytical approach to the Schrödinger-Newton equations, Nonlinearity, 12, 201-216 (1999) · Zbl 0942.35077
[38] Wang, X., On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 53, 229-244 (1993) · Zbl 0795.35118
[39] Willem, M., Minimax Theorems (1996), Berlin: Birkhäuser, Berlin
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.