Multiple solutions to a magnetic nonlinear Choquard equation

By Silvia Cingolani, M. Clapp, Simone Secchi. 2011

258 citations recorded by the source corpus.

AbstractWe consider the stationary nonlinear magnetic Choquard equation $$(- {\rm i}\nabla+ A(x))^{2}u + V (x)u = \left(\frac{1}{|x|^{\alpha}}\ast |u|^{p}\right) |u|^{p-2}u,\quad x\in\mathbb{R}^{N}$$where A is a real-valued vector potential, V is a real-valued scalar potential, N ≥ 3, $${\alpha \in (0, N)}$$ and 2 − (α/N) < p < (2N − α)/(N−2). We assume that both A and V are compatible with the action of some group G of linear isometries of $${\mathbb{R}^{N}}$$ . We establish the existence of multiple complex valued solutions to this equation which satisfy the symmetry condition$$u(gx) = \tau(g)u(x)\quad{\rm for\, all }\ g \in G,\;x \in \mathbb{R}^{N},$$where $${\tau : G \rightarrow \mathbb{S}^{1}}$$ is a given group homomorphism into the unit complex numbers.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/0d71080e5c6641022724767fe11e11e77a491334

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In Applied Mathematics

Citations in the corpus

On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents (2016, background): “the lastdecades. In [34], Lieb provedthe existence anduniqueness, up to translations, of the ground state. Later, in [36], Lions showed the existence of a sequence of radially symmetric solutions. In [21, 37, 38] the authors showed the regularity, positivity and radial symmetry of the ground states ”
Least action nodal solutions for the quadratic Choquard equation (2015, background): “2 MARCO GHIMENTI, VITALY MOROZ, AND JEAN VAN SCHAFTINGEN with a parameter p > 1 attracted interest of mathematicians, see [3, 7, 8, 10,23,26] and further references therein; see also [9, 11] for modifications of (Cp) involving fractional Laplacian.”
Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics (2012, unknown): “ved under the assumption p≥ 2 and [2, 2N N−2]∩ (p, pN α ) ∩( (2p−2)N α+2 , (2p−1)N α+2 )∩ [ (2p−1)N (1.4) N+α ,∞) 6= ∅ respectively by Ma Li and Zhao Lin [20] and by S.Cingolani, M.Clapp and S.Secchi [10]. We prove here the regularity, positivity and radial symmetry of groundstates of Choquard equat”
The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation (2016, background): “ last decades. In [22], Lieb proved the existence and uniqueness, up to translations, of the ground state. Later, in [24], Lions showed the existence of a sequence of radially symmetric solutions. In [16, 25, 26] the authors showed the regularity, positivity and radial symmetry of the ground states ”
Multi-bump solutions for Choquard equation with deepening potential well (2015, result): “ last decades. In [20], Lieb proved the existence and uniqueness, up to translations, of the ground state. Later, in [22], Lions showed the existence of a sequence of radially symmetric solutions. In [12, 25, 26] the authors showed the regularity, positivity and radial symmetry of the ground ∗C. O. ”
Semi-classical states for the Choquard equation (2013, background): “Recently, for ﬁxed ε > 0 and when V converges exponentially to a positive limit at inﬁnity, the existence at least one positive solution to ( ε ) has been proved [16, theorem 1.2; 20].”
Positive and sign changing solutions to a nonlinear Choquard equation (2012, background): “1 in [6] taking A = 0, G = Γ, τ = φ (notice that Z/2 is a subgroup of S(1)) and un ∈ H(1) 0 (Ω) ⊂ H(1)(R ,C).”