Concentration of nodal solutions for semiclassical quadratic Choquard equations
DOI:
https://doi.org/10.58997/ejde.2023.75Abstract
In this article concerns the semiclassical Choquard equation \(-\varepsilon^2 \Delta u +V(x)u = \varepsilon^{-2}( \frac{1}{|\cdot|}* u^2)u\) for \(x \in \mathbb{R}^3\) and small \(\varepsilon\). We establish the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function \(V\), by means of the perturbation method and the method of invariant sets of descending flow.
For more information see https://ejde.math.txstate.edu/Volumes/2023/75/abstr.html
References
N. Ackermann; On a periodic Schro ̈dinger equation with nonlocal superlinear part. Math. Z., 248 (2004), 423–443.
C. O. Alves, H. Luo, M. Yang; Ground state solutions for a class of strongly indefinite Choquard equations. Bull. Malays. Math. Sci. Soc., 43 (2020), 3271–3304.
C. O. Alves, A. B. N ́obrega, M. Yang; Multi-bump solutions for Choquard equation with deepening potential well. Calc. Var. Partial Differential Equations, 55 (2016), 1–28.
J. Byeon, Z. Q. Wang; Standing waves with a critical frequency for nonlinear Schrodinger equations. Arch. Ration. Mech. Anal., 165 (2002), 295–316.
D. Cassani, J. Van Schaftingen, J. Zhang; Ground states for Choquard type equations with Hardy-Littlewood-Sobolev lower critical exponent. Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 1377–1400.
S. Chen, Z. Q. Wang; Localized nodal solutions of higher topological type for semiclassical nonlinear Schro ̈dinger equations. Calc. Var. Partial Differential Equations, 56 (2017), 1–26.
S. Cingolani, K. Tanaka; Semi-classical states for the nonlinear Choquard equations: ex- istence, multiplicity and concentration at a potential well. Rev. Mat. Iberoam., 35 (2019), 1885–1924.
M. Clapp, D. Salazar; Positive and sign changing solutions to a nonlinear Choquard equation. J. Math. Anal. Appl., 407 (2013), 1–15.
F. Gao, E. D. da Silva, M. Yang, J. Zhou; Existence of solutions for critical Choquard equations via the concentration-compactness method. Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 921–954.
F. Gao, M. Yang; On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents. J. Math. Anal. Appl., 448 (2017), 1006–1041.
M. Ghimenti, V. Moroz, J. Van Schaftingen; Least action nodal solutions for the quadratic Choquard equation. Proc. Amer. Math. Soc., 145 (2017), 737–747.
M. Ghimenti, J. Van Schaftingen; Nodal solutions for the Choquard equation. J. Funct. Anal., 271 (2016), 107–135.
C. Gui, H. Guo; On nodal solutions of the nonlinear Choquard equation. Adv. Nonlinear Stud., 19 (4)(2019), 677–691.
L. Guo, T. Hu, S. Peng, W. Shuai; Existence and uniqueness of solutions for Choquard equation involving Hardy-Littlewood-Sobolev critical exponent. Calc. Var. Partial Differential Equations, 58 (2019), 1–34.
R. He, X. Liu; Localized nodal solutions for semiclassical Choquard equations. J. Math. Phys., 62 (2021), 091511.
Z. Huang, J. Yang, W. Yu; Multiple nodal solutions of nonlinear Choquard equations. Electron. J. Differential Equations, 268 (2017), 1–18.
X. Li, S. Ma; Choquard equations with critical nonlinearities. Commun. Contemp. Math., 22 (2020), 1950023, 28.
E. H. Lieb; Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Studies Appl. Math., 57 (1976/77), 93–105.
E. H. Lieb, M. Loss; Analysis, second ed., in Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001.
P. L. Lions; The Choquard equation and related questions. Nonlinear Anal., 4(1980), 1063–1072.
J. Liu, X. Liu, Z. Q. Wang; Sign-changing solutions for coupled nonlinear Schrodinger equations with critical growth. J. Differential Equations, 261 (2016), 7194–7236.
J. Liu, X. Liu, Z. Q. Wang; Multiple mixed states of nodal solutions for nonlinear Schrodinger systems. Calc. Var. Partial Differential Equations, 52 (2015), 565–586.
X. Liu; Localized nodal solutions for system of critical Choquard equations. Commun. Non- linear Sci. Numer. Simul., 121 (2023), 107190.
X. Liu, J. Liu, Z. Q. Wang; Quasilinear elliptic equations via perturbation method. Proc. Amer. Math. Soc., 141 (2013), 253–263.
L. Ma, L. Zhao; Classification of positive solitary solutions of the nonlinear Choquard equa- tion. Arch. Ration. Mech. Anal., 195 (2010), 455–467.
V. Moroz, J. Van Schaftingen; Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J. Funct. Anal., 265 (2013), 153–184.
V. Moroz, J. Van Schaftingen; Groundstates of nonlinear Choquard equations: Hardy- Littlewood-Sobolev critical exponent. Commun. Contemp. Math., 17 (2015), 1550005, 12.
V. Moroz, J. Van Schaftingen; Semi-classical states for the Choquard equation. Calc. Var. Partial Differential Equations, 52 (2015), 199–235.
I. M. Moroz, R. Penrose, P. Tod; Spherically-symmetric solutions of the Schrodinger-Newton equations. Class. Quantum Gravity, 15 (1998), 2733–2742.
S. I. Pekar; Untersuchungen u ̈ber die elektronentheorie der kristalle. De Gruyter, 1954.
D. Qin, V. D. Ra ̆dulescu, X. Tang; Ground states and geometrically distinct solutions for periodic Choquard-Pekar equations. J. Differential Equations, 275 (2021), 652–683.
D. Ruiz, J. Van Schaftingen; Odd symmetry of least energy nodal solutions for the Choquard equation. J. Differential Equations, 264 (2018), 1231–1262.
K. Tintarev, K. H. Fieseler; Concentration compactness. Functional-analytic grounds and applications. Imperial College Press, London, 2007.
B. Zhang, X. Liu; Localized nodal solutions for semiclassical quasilinear Choquard equations with subcritical growth. Electron. J. Differential Equations, 11 (2022), 1–29.
C. L. Xiang; Uniqueness and nondegeneracy of ground states for Choquard equations in three dimensions. Calc. Var. Partial Differential Equations, 55 (2016), 1–25.
C. X. Zhang, Z.-Q. Wang. Concentration of nodal solutions for logarithmic scalar field equations. J. Math. Pures Appl., 135 (2020), 1–25.
Downloads
Published
License
Copyright (c) 2023 Lu Yang, Xiangqing Liu, Jianwen Zhou
This work is licensed under a Creative Commons Attribution 4.0 International License.
