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  1. Calculus of Variations and Partial Differential Equations
  2. Calculus of Variations and Partial Differential Equations : Volume 52
  3. Calculus of Variations and Partial Differential Equations : Volume 52, Issue 1-2, January 2015
  4. Semi-classical states for the Choquard equation
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Calculus of Variations and Partial Differential Equations : Volume 56
Calculus of Variations and Partial Differential Equations : Volume 55
Calculus of Variations and Partial Differential Equations : Volume 54
Calculus of Variations and Partial Differential Equations : Volume 53
Calculus of Variations and Partial Differential Equations : Volume 52
Calculus of Variations and Partial Differential Equations : Volume 52, Issue 3-4, March 2015
Calculus of Variations and Partial Differential Equations : Volume 52, Issue 1-2, January 2015
No neck for Dirac-harmonic maps
Curvature estimates for minimal hypersurfaces via generalized longitude functions
Monge–Ampère equation on exterior domains
On the minimizers of calculus of variations problems in Hilbert spaces
Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian
Concentration-compactness principle and extremal functions for a sharp Trudinger-Moser inequality
A minimal interface problem arising from a two component Bose–Einstein condensate via $$\Gamma $$ -convergence
Semi-classical states for the Choquard equation
Homogenization on arbitrary manifolds
Localization of nonlocal gradients in various topologies
End-to-end construction for the Allen–Cahn equation in the plane
The $$L^\infty $$ optimal transport: infinite cyclical monotonicity and the existence of optimal transport maps
Form-type equations on Kähler manifolds of nonnegative orthogonal bisectional curvature
On the local geometry of maps with c-convex potentials
A new kind of blowing-up solutions for the Brezis-Nirenberg problem
Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case
Calculus of Variations and Partial Differential Equations : Volume 51
Calculus of Variations and Partial Differential Equations : Volume 50
Calculus of Variations and Partial Differential Equations : Volume 49
Calculus of Variations and Partial Differential Equations : Volume 48
Calculus of Variations and Partial Differential Equations : Volume 47
Calculus of Variations and Partial Differential Equations : Volume 46
Calculus of Variations and Partial Differential Equations : Volume 45
Calculus of Variations and Partial Differential Equations : Volume 44
Calculus of Variations and Partial Differential Equations : Volume 43
Calculus of Variations and Partial Differential Equations : Volume 42
Calculus of Variations and Partial Differential Equations : Volume 41
Calculus of Variations and Partial Differential Equations : Volume 40
Calculus of Variations and Partial Differential Equations : Volume 39
Calculus of Variations and Partial Differential Equations : Volume 38
Calculus of Variations and Partial Differential Equations : Volume 37
Calculus of Variations and Partial Differential Equations : Volume 36
Calculus of Variations and Partial Differential Equations : Volume 35
Calculus of Variations and Partial Differential Equations : Volume 34
Calculus of Variations and Partial Differential Equations : Volume 33
Calculus of Variations and Partial Differential Equations : Volume 32
Calculus of Variations and Partial Differential Equations : Volume 31
Calculus of Variations and Partial Differential Equations : Volume 30
Calculus of Variations and Partial Differential Equations : Volume 29
Calculus of Variations and Partial Differential Equations : Volume 28
Calculus of Variations and Partial Differential Equations : Volume 27
Calculus of Variations and Partial Differential Equations : Volume 26
Calculus of Variations and Partial Differential Equations : Volume 25
Calculus of Variations and Partial Differential Equations : Volume 24
Calculus of Variations and Partial Differential Equations : Volume 23
Calculus of Variations and Partial Differential Equations : Volume 22
Calculus of Variations and Partial Differential Equations : Volume 21
Calculus of Variations and Partial Differential Equations : Volume 20
Calculus of Variations and Partial Differential Equations : Volume 19
Calculus of Variations and Partial Differential Equations : Volume 18
Calculus of Variations and Partial Differential Equations : Volume 17
Calculus of Variations and Partial Differential Equations : Volume 16
Calculus of Variations and Partial Differential Equations : Volume 15
Calculus of Variations and Partial Differential Equations : Volume 14
Calculus of Variations and Partial Differential Equations : Volume 13
Calculus of Variations and Partial Differential Equations : Volume 12
Calculus of Variations and Partial Differential Equations : Volume 11
Calculus of Variations and Partial Differential Equations : Volume 10
Calculus of Variations and Partial Differential Equations : Volume 9
Calculus of Variations and Partial Differential Equations : Volume 8
Calculus of Variations and Partial Differential Equations : Volume 7
Calculus of Variations and Partial Differential Equations : Volume 6
Calculus of Variations and Partial Differential Equations : Volume 5

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Semi-classical states for the Choquard equation

Content Provider SpringerLink
Author Moroz, Vitaly Schaftingen, Jean
Copyright Year 2014
Abstract We study the nonlocal equation $$\begin{aligned} -\varepsilon ^2 \Delta u_\varepsilon +V u_\varepsilon = \varepsilon ^{-\alpha }\bigl (I_\alpha *|u_\varepsilon |^p\bigr ) |u_\varepsilon |^{p - 2} u_\varepsilon \quad \text {in }{\mathbb {R}}^N, \end{aligned}$$ where $$N \ge 1$$ , $$\alpha \in (0, N)$$ , $$I_\alpha (x) = A_\alpha /|x |^{N - \alpha }$$ is the Riesz potential and $$\varepsilon > 0$$ is a small parameter. We show that if the external potential $$V \in C ({\mathbb {R}}^N; [0, \infty ))$$ has a local minimum and $$p \in [2, (N + \alpha )/(N - 2)_+)$$ then for all small $$\varepsilon > 0$$ the problem has a family of solutions concentrating to the local minimum of $$V$$ provided that: either $$p>1 + \max (\alpha , \frac{\alpha + 2}{2})/(N - 2)_+$$ , or $$p > 2$$ and $$\liminf _{|x | \rightarrow \infty } V (x) |x |^2 > 0$$ , or $$p = 2$$ and $$\inf _{x \in {\mathbb {R}}^N} V (x) (1 + |x |^{N-\alpha })>0$$ . Our assumptions on the decay of $$V$$ and admissible range of $$p\ge 2$$ are optimal. The proof uses variational methods and a novel nonlocal penalization technique that we develop in this work.
Ending Page 235
Page Count 37
Starting Page 199
File Format PDF
ISSN 09442669
e-ISSN 14320835
Journal Calculus of Variations and Partial Differential Equations
Issue Number 1-2
Volume Number 52
Language English
Publisher Springer Berlin Heidelberg
Publisher Date 2014-03-04
Publisher Place Berlin, Heidelberg
Access Restriction One Nation One Subscription (ONOS)
Subject Keyword Positive solutions Systems Theory, Control Calculus of Variations and Optimal Control; Optimization Analysis Theoretical, Mathematical and Computational Physics Singular perturbations Semilinear elliptic equations Critical exponents NLS-like equations (nonlinear Schrödinger) Asymptotic behavior of solutions Integro-partial differential equations
Content Type Text
Resource Type Article
Subject Applied Mathematics Analysis
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