In this paper, we consider the following magnetic Laplace nonlinear Choquard equation
$ \begin{equation*} -\Delta_A u+V(x)u = (I_{\alpha}*F(|u|))\frac{f(|u|)}{|u|}u, \, \, \text{in}\, \, \mathbb{R}^N, \ \end{equation*} $
where $ u: \mathbb{R}^N\rightarrow C $, $ A: \mathbb{R}^N\rightarrow \mathbb{R}^N $ is a vector potential, $ N\geq 3 $, $ \alpha\, \in\, (N-2, N) $, $ V:\, \mathbb{R}^N \rightarrow \mathbb{R} $ is a scalar potential function and $ I_{\alpha} $ is a Riesz potential of order $ \alpha\, \in\, (N-2, N) $. Under certain assumptions on $ A(x) $, $ V(x) $ and $ f(t) $, we prove that the equation has at least a ground state solution by variational methods.
Citation: Li Zhou, Chuanxi Zhu. Ground state solution for a class of magnetic equation with general convolution nonlinearity[J]. AIMS Mathematics, 2021, 6(8): 9100-9108. doi: 10.3934/math.2021528
In this paper, we consider the following magnetic Laplace nonlinear Choquard equation
$ \begin{equation*} -\Delta_A u+V(x)u = (I_{\alpha}*F(|u|))\frac{f(|u|)}{|u|}u, \, \, \text{in}\, \, \mathbb{R}^N, \ \end{equation*} $
where $ u: \mathbb{R}^N\rightarrow C $, $ A: \mathbb{R}^N\rightarrow \mathbb{R}^N $ is a vector potential, $ N\geq 3 $, $ \alpha\, \in\, (N-2, N) $, $ V:\, \mathbb{R}^N \rightarrow \mathbb{R} $ is a scalar potential function and $ I_{\alpha} $ is a Riesz potential of order $ \alpha\, \in\, (N-2, N) $. Under certain assumptions on $ A(x) $, $ V(x) $ and $ f(t) $, we prove that the equation has at least a ground state solution by variational methods.
[1] | V. Moroz, J. Van Schaftingen, Ground states of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153–184. doi: 10.1016/j.jfa.2013.04.007 |
[2] | V. Moroz, J. Van Schaftingen, Existence of ground states for a class of nonlinear Choquard equations, T. Am. Math. Soc., 367 (2015), 6557–6579. |
[3] | V. Moroz, J. Van Schaftingen, Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains, J. Differ. Equations, 254 (2013), 3089–3145. doi: 10.1016/j.jde.2012.12.019 |
[4] | V. Moroz, J. Van Schaftingen, Ground states of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1550005. doi: 10.1142/S0219199715500054 |
[5] | V. Moroz, J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773–813. doi: 10.1007/s11784-016-0373-1 |
[6] | E. H. Lieb, M. Loss, Analysis, Math. Gazette, 83 (1999), 565–566. |
[7] | M. Ghimenti, J. Van Schaftingen, Nodal solutions for the Choquard equation, J. Funct. Anal., 271 (2016), 107–135. doi: 10.1016/j.jfa.2016.04.019 |
[8] | L. Ma, L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch Rational Mech. Aral., 195 (2010), 455–467. doi: 10.1007/s00205-008-0208-3 |
[9] | D. F. Lü, A note on Kirchhoff-type equations with Hartree-type nonlinearities, Nonlinear Anal.-Theor., 99 (2014), 35–48. doi: 10.1016/j.na.2013.12.022 |
[10] | M. J. Esteban, P. L. Lions, Stationary solutions of a nonlinear Schrödinger equations with an external magnetic field, In: Partial differential equations and the calculus of variations, Boston: Birkhäuser, 1989,401–409. |
[11] | G. Arioli, A. Szulkin, A semilinear Schrödinger equation in the presence of a magnetic field, Arch. Ration. Mech. Anal., 170 (2003), 277–293. doi: 10.1007/s00205-003-0274-5 |
[12] | C. O. Alves, G. M. Figueiredo, M. F. Furtado, Multiple solutions for a nonlinear Schrödinger equation with magnetic fields, Commun. Part. Diff. Eq., 36 (2011), 1565–1586. doi: 10.1080/03605302.2011.593013 |
[13] | M. Clapp, A. Szulkin, Multiple solutions to a nonlinear Schrödinger equation with Aharonov-Bohm magnetic potential, Nonlinear Differ. Equ. Appl., 17 (2010), 229–248. doi: 10.1007/s00030-009-0051-8 |
[14] | C. O. Alves, G. M. Figueiredo, Multiple solutions for a semilinear elliptic equation with critical growth and magnetic field, Milan J. Math., 82 (2014), 389–405. doi: 10.1007/s00032-014-0225-7 |
[15] | C. Ji, V. D. Rädulescu, Multi-bump solutions for the nonlinear magnetic Schrödinger equation with exponential critical growth in $ \mathbb{R}^2$, Manuscripta Math., 164 (2021), 509–542. doi: 10.1007/s00229-020-01195-1 |
[16] | C. Ji, V. D. Rädulescu, Multiplicity and concentration of solutions to the nonlinear magnetic Schrödinger equation, Calc. Var., 59 (2020), 115. doi: 10.1007/s00526-020-01772-y |
[17] | H. M. Nguyen, A. Pinamonti, M. Squassina, E. Vecchi, New characterizations of magnetic Soblev spaces, Adv. Nonlinear Anal., 7 (2018), 227–245. doi: 10.1515/anona-2017-0239 |
[18] | A. Xia, Multiplicity and concentration results for magnetic relativistic Schrödinger equations, Adv. Nonlinear Anal., 9 (2020), 1161–1186. |
[19] | S. Cingolani, M. Clapp, S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63 (2012), 233–248. doi: 10.1007/s00033-011-0166-8 |
[20] | S. Cingolani, M. Clapp, S. Secchi, Intertwining semiclassical solutions to a Schrödinger-Newton system, DCDS-S, 6 (2013), 891–908. |
[21] | S. Cingolani, S. Secchi, M. Squassina, Semi-classical limit for Schrödinger equations solutions with magnetic field and Hartree-type nonlinearities, P. Roy. Soc. Edinb. A, 140A (2010), 973–1009. |
[22] | M. B. Yang, Y. H. Wei, Existence and multiplicity of solutions for nonlinear Schrödinger equations solutions with magnetic field and Hartree type nonlinearities, J. Math. Anal. Appl., 403 (2013), 680–694. doi: 10.1016/j.jmaa.2013.02.062 |
[23] | H. Bueno, G. G. Mamani, G. A. Pereira, Ground state of a magnetic nonlinear Choquard equation, Nonlinear Anal., 181 (2019), 189–199. doi: 10.1016/j.na.2018.11.012 |
[24] | M. Willem, Minimax theorems, In: Progress in nonlinear differential equations and their applications, Boston: Birkhäuser, 1996. |