Compactness and Topological Methods for some Nonlinear Variational Problems of Mathematical Physics

doi:10.1016/S0304-0208(08)71038-7

North-Holland Mathematics Studies

Volume 61, 1982, Pages 17-34

https://doi.org/10.1016/S0304-0208(08)71038-7 Get rights and content

A general method to solve some nonlinear variational problems of Mathematical Physics is illustrated on two examples: the so-called Hartree and Choquard equations. This method is based upon the use of, first, critical point theory and, second, symmetries in order to gain compactness.

References (41)

A. Ambrosetti et al.
Dual variational methods in critical point theory and applications
J. Funct. Anal.
(1973)
N. Bazley et al.
Existence and bounds for critical energies of the Hartree operator
Chem. Phys. Letters
(1974)
M.S. Berger
On the existence and structure of stationary states for a nonlinear Klein-Gordon equation
J. Funct. Anal.
(1972)
M.G. Crandall et al.
Bifurcation from simple eigenvalues
J. Funct. Anal.
(1971)
P.L. Lions
Minimization problems in L¹(IR³)
J. Funct. Anal.
(1981)
C.A. Stuart
An example in nonlinear functional analysis: the Hartree equation
J. Math. Anal. Appl.
(1975)
J.F.G. Auchmuty
Existence of axisymmetric equilibrium fiqures
Arch. Rat. Mech. Anal.
(1977)
J.F.G. Auchmuty et al.
Variational solutions of some nonlinear free boundary problems
Arch. Rat. Mech. Anal.
(1971)
P. Bader
Variational method for the Hartree equation of the Helium atom
Proc. Roy. Soc. Edim.
(1978)
BazleyN. et al.
A branch of positive solutions of nonlinear eigenvalue problems
Manuscripts Math.
(1970)

R. Benguria, H. Brézis, E.H. Lieb, The Thomas-Fermi-von Weizäcker theory of atoms and molecules, to appear in Comm....

H. Berestycki, P.L. Lions, Existence of solutions for nonlinear scalar field equations. I. The ground state, to appear...

H. Berestycki, P.L. Lions, Existence of solutions for nonlinear scalar field equations. II. Existence of infinitely...

H. Berestycki, P.L. Lions, to...

H. Berestycki et al.

Existence of stationary states of nonlinear scalar field equations.

Bifurcation phenomena in Mathematical physics

(1979)

H. Berestycki et al.

Existence d'ondes solitaires dans des problèmes nonlinéaires du type Klein-Gordon

Comptes-Rendus Paris

(1979)

H. Berestycki et al.

Existence of a ground state in nonlinear equations of the type Klein-Gordon.

Variational Inequalities

(1979)

H. Berestycki et al.

Une méthode locale pour l'existence de solutions positives de problèmes semi-linéaires elliptiques dans ℜ^N

J. Anal. Math.

(1980)

H. Berestycki et al.

An O.D.E. approach to the existence of positive solutions for semi-linear problems in ℜ^N

Ind. Univ. Math. J.

(1981)

T. Cazenave P.L. Lions, to...

Cited by (33)

Ergodic Mean-Field Games with aggregation of Choquard-type
2023, Journal of Differential Equations
We consider second-order ergodic Mean-Field Games systems in the whole space $R^{N}$ with coercive potential and aggregating nonlocal coupling, defined in terms of a Riesz interaction kernel. These MFG systems describe Nash equilibria of games with a large population of indistinguishable rational players attracted toward regions where the population is highly distributed. Equilibria solve a system of PDEs where an Hamilton-Jacobi-Bellman equation is combined with a Kolmogorov-Fokker-Planck equation for the mass distribution. Due to the interplay between the strength of the attractive term and the behavior of the diffusive part, we will obtain three different regimes for the existence and non existence of classical solutions to the MFG system. By means of a Pohozaev-type identity, we prove nonexistence of regular solutions to the MFG system without potential in the Hardy-Littlewood-Sobolev-supercritical regime. On the other hand, using a fixed point argument, we show existence of classical solutions in the Hardy-Littlewood-Sobolev-subcritical regime at least for masses smaller than a given threshold value. In the mass-subcritical regime we show that actually this threshold can be taken to be +∞.
Blow-up of ground states of fractional Choquard equations
2022, Nonlinear Analysis, Theory, Methods and Applications
Semiclassical states for critical Choquard equations
2021, Journal of Mathematical Analysis and Applications
Remarks about a fractional Choquard equation: Ground state, regularity and polynomial decay
2017, Nonlinear Analysis, Theory, Methods and Applications
Instability of standing waves for inhomogeneous Hartree equations
2016, Journal of Mathematical Analysis and Applications
Existence and nonexistence results for a class of Hamiltonian Choquard-type elliptic systems with lower critical growth on ℝ<sup>2</sup>
2022, Proceedings of the Royal Society of Edinburgh Section A: Mathematics

View all citing articles on Scopus

View full text

Compactness and Topological Methods for some Nonlinear Variational Problems of Mathematical Physics

J. Funct. Anal.

Chem. Phys. Letters

J. Funct. Anal.

J. Funct. Anal.

J. Funct. Anal.

J. Math. Anal. Appl.

Existence of axisymmetric equilibrium fiqures

Arch. Rat. Mech. Anal.

Variational solutions of some nonlinear free boundary problems

Arch. Rat. Mech. Anal.

Variational method for the Hartree equation of the Helium atom

Proc. Roy. Soc. Edim.

A branch of positive solutions of nonlinear eigenvalue problems

Manuscripts Math.

Existence of stationary states of nonlinear scalar field equations.

Bifurcation phenomena in Mathematical physics

Existence d'ondes solitaires dans des problèmes nonlinéaires du type Klein-Gordon

Comptes-Rendus Paris

Existence of a ground state in nonlinear equations of the type Klein-Gordon.

Variational Inequalities

Une méthode locale pour l'existence de solutions positives de problèmes semi-linéaires elliptiques dans ℜN

J. Anal. Math.

An O.D.E. approach to the existence of positive solutions for semi-linear problems in ℜN

Ind. Univ. Math. J.

Une méthode locale pour l'existence de solutions positives de problèmes semi-linéaires elliptiques dans ℜ^N

An O.D.E. approach to the existence of positive solutions for semi-linear problems in ℜ^N