Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire, ISSN 0294-1449, Volume 9, Issue 3, pp. 281 - 304

Let p=[formula presented], N ≧ 3 be the limiting Sobolev exponent and Ω ⊂ ℝ open bounded set. We show that for f ∈ H satisfying a suitable condition and f ≠ 0,...

Semilinear elliptic equations | critical Sobolev exponent | MATHEMATICS, APPLIED | SEMILINEAR ELLIPTIC EQUATIONS | CRITICAL SOBOLEV EXPONENT | POSITIVE SOLUTIONS

Semilinear elliptic equations | critical Sobolev exponent | MATHEMATICS, APPLIED | SEMILINEAR ELLIPTIC EQUATIONS | CRITICAL SOBOLEV EXPONENT | POSITIVE SOLUTIONS

Journal Article

Mediterranean Journal of Mathematics, ISSN 1660-5446, 4/2019, Volume 16, Issue 2, pp. 1 - 27

In this work, we have proved a Hardy–Littlewood–Sobolev inequality for variable exponents. After that, we use this inequality together with the variational...

quasilinear elliptic equations | 35J62 | 35A15 | Variational methods | 35J60 | Choquard equation | Mathematics, general | Mathematics | nonlinear elliptic equations | EXISTENCE | MATHEMATICS, APPLIED | MULTIPLICITY | P(X)-LAPLACIAN EQUATIONS | POTENTIALS | CONVOLUTION | MATHEMATICS | R-N | REGULARITY | CRITICAL GROWTH | LEBESGUE SPACES | OPERATORS | Equality

quasilinear elliptic equations | 35J62 | 35A15 | Variational methods | 35J60 | Choquard equation | Mathematics, general | Mathematics | nonlinear elliptic equations | EXISTENCE | MATHEMATICS, APPLIED | MULTIPLICITY | P(X)-LAPLACIAN EQUATIONS | POTENTIALS | CONVOLUTION | MATHEMATICS | R-N | REGULARITY | CRITICAL GROWTH | LEBESGUE SPACES | OPERATORS | Equality

Journal Article

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Groundstates of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent

Communications in Contemporary Mathematics, ISSN 0219-1997, 10/2015, Volume 17, Issue 5, pp. 1550005 - 1550001

We consider nonlinear Choquard equation $$ - \Delta u + V u = (I_\alpha \ast \vert u\vert^{\frac{\alpha}{N}+1}) \vert u\vert^{\frac{\alpha}{N}-1} u\quad {\rm...

Hartree equation | Choquard equation | strict inequality | Hardy-Littlewood-Sobolev inequality | concentration at infinity | concentration-compactness | nonlinear Schrödinger equation | nonlocal problem | Riesz potential | lower critical exponent | MATHEMATICS, APPLIED | DECAY | nonlinear Schrodinger equation | MATHEMATICS | Exponents | Mathematical analysis | Inequalities | Texts | Minimization | Nonlinearity | Optimization | Mathematics - Analysis of PDEs

Hartree equation | Choquard equation | strict inequality | Hardy-Littlewood-Sobolev inequality | concentration at infinity | concentration-compactness | nonlinear Schrödinger equation | nonlocal problem | Riesz potential | lower critical exponent | MATHEMATICS, APPLIED | DECAY | nonlinear Schrodinger equation | MATHEMATICS | Exponents | Mathematical analysis | Inequalities | Texts | Minimization | Nonlinearity | Optimization | Mathematics - Analysis of PDEs

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 09/2019, Volume 78, Issue 6, pp. 2063 - 2082

In this paper, we consider the following fractional Kirchhoff-type equation: a+b∫RN∫RN|u(x)−u(y)|2|x−y|N+2sdxdy(−Δ)su=Iα∗|u|2h,α∗|u|2h,α∗−2u,inRN,where N⩾3,...

Moser iteration scheme | Fractional Kirchhoff-type equation | Hardy–Littlewood–Sobolev critical exponent | EXISTENCE | MATHEMATICS, APPLIED | MULTIPLICITY | POSITIVE SOLUTIONS | Hardy-Littlewood-Sobolev critical exponent

Moser iteration scheme | Fractional Kirchhoff-type equation | Hardy–Littlewood–Sobolev critical exponent | EXISTENCE | MATHEMATICS, APPLIED | MULTIPLICITY | POSITIVE SOLUTIONS | Hardy-Littlewood-Sobolev critical exponent

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 09/2016, Volume 261, Issue 6, pp. 3061 - 3106

In this paper, we study the existence of ground state solutions for the nonlinear fractional Schrödinger–Poisson system with critical Sobolev...

Fractional Schrödinger–Poisson system | Variational methods | Palais–Smale condition | Pohozaev identity | Palais-Smale condition | POSITIVE SOLUTIONS | CALCULUS | CONCENTRATION-COMPACTNESS PRINCIPLE | MAXWELL EQUATIONS | Fractional Schrodinger-Poisson system | GUIDE | LAPLACIAN | MATHEMATICS | R-N | CRITICAL GROWTH

Fractional Schrödinger–Poisson system | Variational methods | Palais–Smale condition | Pohozaev identity | Palais-Smale condition | POSITIVE SOLUTIONS | CALCULUS | CONCENTRATION-COMPACTNESS PRINCIPLE | MAXWELL EQUATIONS | Fractional Schrodinger-Poisson system | GUIDE | LAPLACIAN | MATHEMATICS | R-N | CRITICAL GROWTH

Journal Article

Journal of Statistical Physics, ISSN 0022-4715, 12/2009, Volume 137, Issue 5, pp. 857 - 877

After a review of the history and an assessment of the current status of the subject, we present light-scattering data to determine the critical...

Physical Chemistry | Theoretical, Mathematical and Computational Physics | Light scattering | Critical correlation function | Gravity effects | Quantum Physics | Statistical Physics, Dynamical Systems and Complexity | Isobutyric acid + water | Physics | Critical exponents | Isobutyric acid + water | RENORMALIZATION-GROUP THEORY | DEUTERATED ISOBUTYRIC ACID | IONIC BINARY-MIXTURE | Isobutyric acid plus water | VAPOR CRITICAL-POINT | PHYSICS, MATHEMATICAL | TERNARY-SYSTEM 3-METHYLPYRIDINE/WATER/NABR | UNIVERSAL AMPLITUDE COMBINATIONS | LIQUID CRITICAL-POINT | EQUILIBRIUM CRITICAL PHENOMENA | CRITICAL SOLUTION POINT | NONIONIC MICELLAR-SOLUTIONS

Physical Chemistry | Theoretical, Mathematical and Computational Physics | Light scattering | Critical correlation function | Gravity effects | Quantum Physics | Statistical Physics, Dynamical Systems and Complexity | Isobutyric acid + water | Physics | Critical exponents | Isobutyric acid + water | RENORMALIZATION-GROUP THEORY | DEUTERATED ISOBUTYRIC ACID | IONIC BINARY-MIXTURE | Isobutyric acid plus water | VAPOR CRITICAL-POINT | PHYSICS, MATHEMATICAL | TERNARY-SYSTEM 3-METHYLPYRIDINE/WATER/NABR | UNIVERSAL AMPLITUDE COMBINATIONS | LIQUID CRITICAL-POINT | EQUILIBRIUM CRITICAL PHENOMENA | CRITICAL SOLUTION POINT | NONIONIC MICELLAR-SOLUTIONS

Journal Article

Polymer, ISSN 0032-3861, 02/2016, Volume 84, pp. 275 - 285

A soluble polyimide (6FDA-TFDB) was synthesized and its properties in solution were investigated. The relationship between specific viscosity and concentration...

Dipole–dipole interaction | Scaling exponent | Critical concentration | Soluble polyimide | Dipole-dipole interaction | ISOMERIZED POLYIMIDES | POLYMER SCIENCE | CHAIN CONFORMATION | BEHAVIOR | VISCOSITY | LOCAL RIGIDITY | DEPENDENCE | Polyimides | Solvents | Dilution | Exponents | Rheometers | Mathematical models | Deviation | Hydrogen bonding | Polyimide resins

Dipole–dipole interaction | Scaling exponent | Critical concentration | Soluble polyimide | Dipole-dipole interaction | ISOMERIZED POLYIMIDES | POLYMER SCIENCE | CHAIN CONFORMATION | BEHAVIOR | VISCOSITY | LOCAL RIGIDITY | DEPENDENCE | Polyimides | Solvents | Dilution | Exponents | Rheometers | Mathematical models | Deviation | Hydrogen bonding | Polyimide resins

Journal Article

Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 09/2019, Volume 42, Issue 14, pp. 4815 - 4838

In this paper, we consider the following Schrödinger‐Poisson system: −Δu+λϕ|u|2α∗−2u=∫R3|u|2β∗|x−y|3−βdy|u|2β∗−2u,inR3,(−Δ)α2ϕ=Aα−1|u|2α∗,inR3, where...

Hardy‐Littlewood‐Sobolev critical exponent | Schrödinger‐Poisson system | Kelvin transformation | Moser iteration | Hardy-Littlewood-Sobolev critical exponent | Schrödinger-Poisson system | EXISTENCE | Schrodinger-Poisson system | MATHEMATICS, APPLIED | EQUATIONS | GROUND-STATE SOLUTIONS | Perturbation methods

Hardy‐Littlewood‐Sobolev critical exponent | Schrödinger‐Poisson system | Kelvin transformation | Moser iteration | Hardy-Littlewood-Sobolev critical exponent | Schrödinger-Poisson system | EXISTENCE | Schrodinger-Poisson system | MATHEMATICS, APPLIED | EQUATIONS | GROUND-STATE SOLUTIONS | Perturbation methods

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 2015, Volume 429, Issue 2, pp. 1153 - 1172

In this paper, we study the following nonlinear Kirchhoff-type equation where is nonnegative. By using the variational method, we obtain the existence of...

Kirchhoff-type equation | Zero mass | Critical exponent | Variational methods | EXISTENCE | MATHEMATICS, APPLIED | STATES | MULTIPLICITY | CRITICAL NONLINEARITY | MATHEMATICS | CRITICAL GROWTH | R-3 | CRITICAL SOBOLEV EXPONENTS | ELLIPTIC-EQUATIONS

Kirchhoff-type equation | Zero mass | Critical exponent | Variational methods | EXISTENCE | MATHEMATICS, APPLIED | STATES | MULTIPLICITY | CRITICAL NONLINEARITY | MATHEMATICS | CRITICAL GROWTH | R-3 | CRITICAL SOBOLEV EXPONENTS | ELLIPTIC-EQUATIONS

Journal Article

Nonlinear Analysis: Real World Applications, ISSN 1468-1218, 12/2016, Volume 32, pp. 198 - 212

Existence of one positive solution of the generalized Schrödinger–Poisson system {−Δu+V(x)u−K(x)ϕ|u|3u=f(x,u)inR3,−Δϕ=K(x)|u|5inR3, where V,K,f are...

Asymptotically periodic | Concentration-compactness principle | Critical exponent | Mountain pass theorem | Generalized Schrödinger–Poisson system | Generalized Schrödinger-Poisson system | EXISTENCE | MATHEMATICS, APPLIED | SPHERES | MAXWELL EQUATIONS | Generalized Schrodinger-Poisson system | PRINCIPLE | GROUND-STATE SOLUTIONS | BOUND-STATES | R-3 | ELLIPTIC-EQUATIONS | Mountains | Theorems | Periodic functions | Exponents | Existence theorems | Asymptotic properties | Images | Nonlinearity

Asymptotically periodic | Concentration-compactness principle | Critical exponent | Mountain pass theorem | Generalized Schrödinger–Poisson system | Generalized Schrödinger-Poisson system | EXISTENCE | MATHEMATICS, APPLIED | SPHERES | MAXWELL EQUATIONS | Generalized Schrodinger-Poisson system | PRINCIPLE | GROUND-STATE SOLUTIONS | BOUND-STATES | R-3 | ELLIPTIC-EQUATIONS | Mountains | Theorems | Periodic functions | Exponents | Existence theorems | Asymptotic properties | Images | Nonlinearity

Journal Article

Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, 06/2017, Volume 50, Issue 26, p. 264003

We implement a scale-free version of the pivot algorithm and use it to sample pairs of three-dimensional self-avoiding walks, for the purpose of efficiently...

pivot algorithm | self-avoiding walk | Monte Carlo | critical exponent

pivot algorithm | self-avoiding walk | Monte Carlo | critical exponent

Journal Article

Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 05/2019, Volume 42, Issue 7, pp. 2417 - 2430

In this paper, the existence and multiplicity of positive solutions is established for Schrödinger‐Poisson system of the form...

Schrödinger‐poisson system | Nehari method | singularity | critical exponent | Schrödinger-poisson system | MATHEMATICS, APPLIED | BOUND-STATES | EQUATIONS | GROUND-STATE SOLUTIONS | Schrodinger-poisson system

Schrödinger‐poisson system | Nehari method | singularity | critical exponent | Schrödinger-poisson system | MATHEMATICS, APPLIED | BOUND-STATES | EQUATIONS | GROUND-STATE SOLUTIONS | Schrodinger-poisson system

Journal Article

Nonlinear Analysis, ISSN 0362-546X, 2009, Volume 70, Issue 6, pp. 2150 - 2164

In this paper we establish the existence of a positive solution of the Schrödinger–Poisson equations with a critical Sobolev exponent. The methods used here...

Pohoz˘aev identity | Concentration–compactness principle | Critical growth | Variational methods | Pohoz ̌aev identity | Concentration-compactness principle | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | STATES | Pohozaev identity

Pohoz˘aev identity | Concentration–compactness principle | Critical growth | Variational methods | Pohoz ̌aev identity | Concentration-compactness principle | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | STATES | Pohozaev identity

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 04/2017, Volume 448, Issue 2, pp. 1006 - 1041

We consider the following nonlinear Choquard equation with Dirichlet boundary condition−Δu=(∫Ω|u|2μ⁎|x−y|μdy)|u|2μ⁎−2u+λf(u)inΩ, where Ω is a smooth bounded...

Concave and convex nonlinearities | Brezis–Nirenberg problem | Choquard equation | Hardy–Littlewood–Sobolev critical exponent | EXISTENCE | CONVEX NONLINEARITIES | MATHEMATICS, APPLIED | MULTIPLICITY | POSITIVE SOLUTIONS | SCHRODINGER-EQUATION | GROUND-STATE SOLUTIONS | Brezis-Nirenberg problem | MATHEMATICS | ELLIPTIC PROBLEMS | OPERATOR | CONCAVE | Hardy-Littlewood-Sobolev critical exponent

Concave and convex nonlinearities | Brezis–Nirenberg problem | Choquard equation | Hardy–Littlewood–Sobolev critical exponent | EXISTENCE | CONVEX NONLINEARITIES | MATHEMATICS, APPLIED | MULTIPLICITY | POSITIVE SOLUTIONS | SCHRODINGER-EQUATION | GROUND-STATE SOLUTIONS | Brezis-Nirenberg problem | MATHEMATICS | ELLIPTIC PROBLEMS | OPERATOR | CONCAVE | Hardy-Littlewood-Sobolev critical exponent

Journal Article

Physical Review B - Condensed Matter and Materials Physics, ISSN 1098-0121, 06/2015, Volume 91, Issue 21

Relations between critical exponents, or scaling laws, at both continuous and discontinuous quantum phase transitions are derived and discussed. In general...

PHYSICS, CONDENSED MATTER | PHYSICS, APPLIED | CRITICAL-BEHAVIOR | MATERIALS SCIENCE, MULTIDISCIPLINARY | CRITICAL-POINTS | Exponents | Phase transformations | Condensed matter | Asymptotic properties | Scaling laws | Inequalities | Ferromagnetism | Dynamical systems | Physics - Statistical Mechanics

PHYSICS, CONDENSED MATTER | PHYSICS, APPLIED | CRITICAL-BEHAVIOR | MATERIALS SCIENCE, MULTIDISCIPLINARY | CRITICAL-POINTS | Exponents | Phase transformations | Condensed matter | Asymptotic properties | Scaling laws | Inequalities | Ferromagnetism | Dynamical systems | Physics - Statistical Mechanics

Journal Article

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Positive solutions for a Schrödinger-Poisson system with singularity and critical exponent

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 03/2020, Volume 483, Issue 2, p. 123647

We study multiplicity of positive solutions for a class of Schrödinger-Poisson system with singularity and critical exponent, and obtain two positive solutions...

Schrödinger-Poisson systems | Critical exponent | Singular nonlinearity | Perturbation approach | EXISTENCE | MATHEMATICS, APPLIED | KLEIN-GORDON-MAXWELL | Schrodinger-Poisson systems | EQUATIONS | GROUND-STATE SOLUTIONS | MATHEMATICS | SOLITARY WAVES | BOUND-STATES | CONCAVE

Schrödinger-Poisson systems | Critical exponent | Singular nonlinearity | Perturbation approach | EXISTENCE | MATHEMATICS, APPLIED | KLEIN-GORDON-MAXWELL | Schrodinger-Poisson systems | EQUATIONS | GROUND-STATE SOLUTIONS | MATHEMATICS | SOLITARY WAVES | BOUND-STATES | CONCAVE

Journal Article

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The Ising magnetization exponent on {Mathematical expression} is {Mathematical expression}

Probability Theory and Related Fields, ISSN 0178-8051, 2013, Volume 160, Issue 1-2, pp. 1 - 13

We prove that for the Ising model defined on the plane at the average magnetization under an external magnetic field behaves exactly like The proof, which is...

Mathematics Subject Classification: 82B20, 82B27, 60K35 | TRANSITION | STATISTICS & PROBABILITY | INEQUALITIES | MODEL | PERCOLATION | CRITICAL-POINT | Magnetization | Exponents | Lattices | Inequalities | Ising model | Texts | Percolation | Magnetic fields

Mathematics Subject Classification: 82B20, 82B27, 60K35 | TRANSITION | STATISTICS & PROBABILITY | INEQUALITIES | MODEL | PERCOLATION | CRITICAL-POINT | Magnetization | Exponents | Lattices | Inequalities | Ising model | Texts | Percolation | Magnetic fields

Journal Article