Nonlinear Analysis, ISSN 0362-546X, 05/2017, Volume 155, pp. 52 - 64

In this paper we consider the nonlinear fractional logarithmic Schrödinger equation. By using a compactness method, we construct a unique global solution of...

Standing waves | Fractional logarithmic Schrödinger equation | Stability | MATHEMATICS | MATHEMATICS, APPLIED | HARTREE TYPE NONLINEARITY | Fractional logarithmic Schrodinger equation | FUNCTIONALS

Standing waves | Fractional logarithmic Schrödinger equation | Stability | MATHEMATICS | MATHEMATICS, APPLIED | HARTREE TYPE NONLINEARITY | Fractional logarithmic Schrodinger equation | FUNCTIONALS

Journal Article

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Full Text
On the Cauchy Problem of Fractional Schrödinger Equation with Hartree Type Nonlinearity

Funkcialaj Ekvacioj, ISSN 0532-8721, 2013, Volume 56, Issue 2, pp. 193 - 224

We study the Cauchy problem for the fractional Schrödinger equation i∂tu = (m2−Δ)α/2u + F(u) in R1+n, where n ≥ 1, m ≥ 0, 1 < α < 2, and F stands for the...

Fractional Schrödinger equation | Strichartz estimates | Finite time blowup | Hartree type nonlinearity | Fractional schrödinger equation | MATHEMATICS | MATHEMATICS, APPLIED | Fractional Schrodinger equation

Fractional Schrödinger equation | Strichartz estimates | Finite time blowup | Hartree type nonlinearity | Fractional schrödinger equation | MATHEMATICS | MATHEMATICS, APPLIED | Fractional Schrodinger equation

Journal Article

Advanced Nonlinear Studies, ISSN 1536-1365, 2019, Volume 19, Issue 4, pp. 779 - 795

In this paper, we consider the Kirchhoff equation with Hartree-type nonlinearity {-(epsilon(2) a + epsilon b integral(R3)vertical bar del u vertical bar(2) dx...

Concentration | Hartree-type Nonlinearity | Kirchhoff Equation | Variational Methods | Multiplicity | SCHRODINGER-EQUATIONS | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | BOUND-STATES | REGULARITY | POSITIVE SOLUTIONS

Concentration | Hartree-type Nonlinearity | Kirchhoff Equation | Variational Methods | Multiplicity | SCHRODINGER-EQUATIONS | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | BOUND-STATES | REGULARITY | POSITIVE SOLUTIONS

Journal Article

Boundary Value Problems, ISSN 1687-2762, 12/2018, Volume 2018, Issue 1, pp. 1 - 15

We use the non-Nehari manifold method to deal with the system {−Δu+V(x)u+ϕu=(∫R3Q(y)F(u(y))|x−y|μdy)Q(x)f(u(x)),x∈R3,−Δϕ=u2,u∈H1(R3), $$...

Asymptotically periodic | Ordinary Differential Equations | Schrödinger–Poisson system | Analysis | Difference and Functional Equations | Approximations and Expansions | Hartree-type nonlinearity | Mathematics, general | Mathematics | Partial Differential Equations | Ground state solutions | EXISTENCE | MATHEMATICS | Schrodinger-Poisson system | MATHEMATICS, APPLIED | MULTIPLICITY | EQUATIONS | GORDON-MAXWELL SYSTEMS | Asymptotic properties | Ground state

Asymptotically periodic | Ordinary Differential Equations | Schrödinger–Poisson system | Analysis | Difference and Functional Equations | Approximations and Expansions | Hartree-type nonlinearity | Mathematics, general | Mathematics | Partial Differential Equations | Ground state solutions | EXISTENCE | MATHEMATICS | Schrodinger-Poisson system | MATHEMATICS, APPLIED | MULTIPLICITY | EQUATIONS | GORDON-MAXWELL SYSTEMS | Asymptotic properties | Ground state

Journal Article

Nonlinear Analysis, ISSN 0362-546X, 2007, Volume 66, Issue 8, pp. 1770 - 1781

We study the inverse scattering problem for the nonlinear Schrödinger equation and for the nonlinear Klein–Gordon equation with the generalized Hartree type...

Inverse scattering | Hartree type nonlinearity | Scattering | CUBIC CONVOLUTION NONLINEARITY | MATHEMATICS | WAVE | MATHEMATICS, APPLIED | scattering | RECONSTRUCTION | LINE | inverse scattering | CAUCHY-PROBLEM | UNIQUENESS

Inverse scattering | Hartree type nonlinearity | Scattering | CUBIC CONVOLUTION NONLINEARITY | MATHEMATICS | WAVE | MATHEMATICS, APPLIED | scattering | RECONSTRUCTION | LINE | inverse scattering | CAUCHY-PROBLEM | UNIQUENESS

Journal Article

Applied Mathematics Letters, ISSN 0893-9659, 07/2018, Volume 81, pp. 21 - 26

In this paper, we study the following Schrödinger equation −Δu+Vλ(x)u+μϕ|u|p−2u=f(x,u)+β(x)|u|ν−2u, in R3,(−Δ)α2ϕ=μ|u|p, in R3,where μ≥0 is a parameter,...

Hartree-type nonlinearity | Variational methods | Schrödinger equations | Sign-changing potential | MATHEMATICS, APPLIED | POISSON SYSTEMS | Schrodinger equations

Hartree-type nonlinearity | Variational methods | Schrödinger equations | Sign-changing potential | MATHEMATICS, APPLIED | POISSON SYSTEMS | Schrodinger equations

Journal Article

Journal of Fixed Point Theory and Applications, ISSN 1661-7738, 3/2017, Volume 19, Issue 1, pp. 773 - 813

We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations $$\begin{aligned} -\Delta u + V(x)u =...

Pekar polaron | Mathematical Methods in Physics | focusing Hartree equation | attractive nonlocal interaction | Choquard equation | Analysis | Mathematics, general | Schrödinger–Newton equation | Mathematics | 35Q55 (35R09, 35J91) | Riesz potential | MATHEMATICS, APPLIED | NONLINEAR SCHRODINGER-EQUATIONS | SCALAR FIELD-EQUATIONS | POSITIVE SOLUTIONS | Schrodinger-Newton equation | FRACTIONAL INTEGRALS | MATHEMATICS | HARDY-LITTLEWOOD-SOBOLEV | MAGNETIC-FIELD | GROUND-STATE ENERGY | BLOW-UP SOLUTIONS | HARTREE-TYPE NONLINEARITIES | GENERAL NONLINEARITY | Differential equations | Mathematics - Analysis of PDEs

Pekar polaron | Mathematical Methods in Physics | focusing Hartree equation | attractive nonlocal interaction | Choquard equation | Analysis | Mathematics, general | Schrödinger–Newton equation | Mathematics | 35Q55 (35R09, 35J91) | Riesz potential | MATHEMATICS, APPLIED | NONLINEAR SCHRODINGER-EQUATIONS | SCALAR FIELD-EQUATIONS | POSITIVE SOLUTIONS | Schrodinger-Newton equation | FRACTIONAL INTEGRALS | MATHEMATICS | HARDY-LITTLEWOOD-SOBOLEV | MAGNETIC-FIELD | GROUND-STATE ENERGY | BLOW-UP SOLUTIONS | HARTREE-TYPE NONLINEARITIES | GENERAL NONLINEARITY | Differential equations | Mathematics - Analysis of PDEs

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 04/2017, Volume 448, Issue 1, pp. 60 - 80

In this paper, we discuss the existence and the concentration of sign-changing solutions to a class of Kirchhoff-type systems with Hartree-type nonlinearity in...

Hartree-type nonlinearity | Sign-changing solution | Kirchhoff-type system | Concentration | MATHEMATICS, APPLIED | MULTIPLICITY | GROUND-STATE | POSITIVE SOLUTIONS | NONTRIVIAL SOLUTIONS | EQUATIONS | HIGH-ENERGY SOLUTIONS | SCHRODINGER-POISSON SYSTEM | MATHEMATICS | R-N | R-3

Hartree-type nonlinearity | Sign-changing solution | Kirchhoff-type system | Concentration | MATHEMATICS, APPLIED | MULTIPLICITY | GROUND-STATE | POSITIVE SOLUTIONS | NONTRIVIAL SOLUTIONS | EQUATIONS | HIGH-ENERGY SOLUTIONS | SCHRODINGER-POISSON SYSTEM | MATHEMATICS | R-N | R-3

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 05/2019, Volume 473, Issue 1, pp. 587 - 608

In this paper we study the Kirchhoff equation with Hartree-type nonlinearities−(a+b∫R3|∇u|2)Δu+V(x)u=(Iα⁎|u|p)|u|p−2uin R3, where a,b>0, V:R3→R is a potential...

Ground state | Kirchhoff equation | Hartree-type nonlinearities | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | CHOQUARD-EQUATIONS | NODAL SOLUTIONS | DECAY | BEHAVIOR

Ground state | Kirchhoff equation | Hartree-type nonlinearities | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | CHOQUARD-EQUATIONS | NODAL SOLUTIONS | DECAY | BEHAVIOR

Journal Article

Nonlinear Analysis, ISSN 0362-546X, 04/2014, Volume 99, pp. 35 - 48

We consider the following Kirchhoff-type equation in R3−(a+b∫R3|∇u|2dx)Δu+(1+μg(x))u=(1|x|α∗|u|p)|u|p−2u, where a>0, b≥0 are constants, α∈(0,3), p∈(2,6−α), μ>0...

Kirchhoff-type equation | Hartree-type nonlinearity | Variational methods | Ground state solution | SCHRODINGER-EQUATIONS | EXISTENCE | MATHEMATICS, APPLIED | MULTIPLICITY | POSITIVE SOLUTIONS | NONTRIVIAL SOLUTIONS | MATHEMATICS | R-N | MAGNETIC-FIELD | Mathematical analysis | Nonlinearity | Manifolds | Texts | Ground state | Constants

Kirchhoff-type equation | Hartree-type nonlinearity | Variational methods | Ground state solution | SCHRODINGER-EQUATIONS | EXISTENCE | MATHEMATICS, APPLIED | MULTIPLICITY | POSITIVE SOLUTIONS | NONTRIVIAL SOLUTIONS | MATHEMATICS | R-N | MAGNETIC-FIELD | Mathematical analysis | Nonlinearity | Manifolds | Texts | Ground state | Constants

Journal Article

PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS, ISSN 0308-2105, 08/2019, Volume 149, Issue 4, pp. 979 - 994

In this paper, we are concerned with the following bi-harmonic equation with Hartree type nonlinearity (P-gamma) Delta(2)u = (1/vertical bar x vertical bar(8)...

SCHRODINGER-EQUATIONS | LOCAL BEHAVIOR | MATHEMATICS, APPLIED | SEMILINEAR ELLIPTIC-EQUATIONS | CONCENTRATION-COMPACTNESS PRINCIPLE | GLOBAL WELL-POSEDNESS | SOBOLEV | UNIQUENESS | MATHEMATICS | DIRICHLET BOUNDARY-CONDITIONS | Liouville type theorems | SYMMETRY | bi-harmonic | Hartree type nonlinearity | methods of moving planes | radial symmetry | nonnegative solutions | SCATTERING | Nonlinearity

SCHRODINGER-EQUATIONS | LOCAL BEHAVIOR | MATHEMATICS, APPLIED | SEMILINEAR ELLIPTIC-EQUATIONS | CONCENTRATION-COMPACTNESS PRINCIPLE | GLOBAL WELL-POSEDNESS | SOBOLEV | UNIQUENESS | MATHEMATICS | DIRICHLET BOUNDARY-CONDITIONS | Liouville type theorems | SYMMETRY | bi-harmonic | Hartree type nonlinearity | methods of moving planes | radial symmetry | nonnegative solutions | SCATTERING | Nonlinearity

Journal Article

Applicable Analysis, ISSN 0003-6811, 01/2018, Volume 97, Issue 2, pp. 255 - 273

In the present paper we deal with a nonlinear fractional Schrödinger equation in of the form where is a parameter, , , V(x) is a continuous potential, and is...

Hartree-type nonlinearity | Fractional Schrödinger equation | existence and multiplicity | concentration behavior | EXISTENCE | LAPLACIAN | ELLIPTIC PROBLEMS | MATHEMATICS, APPLIED | Fractional Schrodinger equation | CHOQUARD EQUATION | MULTIPLE POSITIVE SOLUTIONS | Nonlinearity | Nonlinear equations | Schroedinger equation | Variational methods | Formulas (mathematics)

Hartree-type nonlinearity | Fractional Schrödinger equation | existence and multiplicity | concentration behavior | EXISTENCE | LAPLACIAN | ELLIPTIC PROBLEMS | MATHEMATICS, APPLIED | Fractional Schrodinger equation | CHOQUARD EQUATION | MULTIPLE POSITIVE SOLUTIONS | Nonlinearity | Nonlinear equations | Schroedinger equation | Variational methods | Formulas (mathematics)

Journal Article

Zeitschrift für angewandte Mathematik und Physik, ISSN 0044-2275, 12/2018, Volume 69, Issue 6, pp. 1 - 20

A Kirchhoff-type fractional elliptic system with Hartree-type nonlinearity is proposed to provide a unified framework for well-known nonlinear Schrödinger...

35J20 | 35J62 | 35J60 | 35Q55 | Variational method | Concentration | Theoretical and Applied Mechanics | 35B09 | Engineering | Mathematical Methods in Physics | Ground state solution | Hartree-type nonlinearity | Kirchhoff-type system | MATHEMATICS, APPLIED | KLEIN-GORDON-MAXWELL | MULTIPLICITY | POSITIVE SOLUTIONS | SEMICLASSICAL STATES | EQUATIONS | SCHRODINGER-POISSON SYSTEM | WAVES | BOUND-STATES | SIGN-CHANGING SOLUTIONS

35J20 | 35J62 | 35J60 | 35Q55 | Variational method | Concentration | Theoretical and Applied Mechanics | 35B09 | Engineering | Mathematical Methods in Physics | Ground state solution | Hartree-type nonlinearity | Kirchhoff-type system | MATHEMATICS, APPLIED | KLEIN-GORDON-MAXWELL | MULTIPLICITY | POSITIVE SOLUTIONS | SEMICLASSICAL STATES | EQUATIONS | SCHRODINGER-POISSON SYSTEM | WAVES | BOUND-STATES | SIGN-CHANGING SOLUTIONS

Journal Article

Mediterranean Journal of Mathematics, ISSN 1660-5446, 6/2018, Volume 15, Issue 3, pp. 1 - 17

This paper is concerned with the following Kirchhoff-type equations: $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\big (a+b\int _{\mathbb...

35B38 | Variational methods | 35J60 | Hartree-type nonlinearity | Mathematics, general | Symmetric mountain pass theorem | Mathematics | Kirchhoff equations | Sign-changing potential | EXISTENCE | MATHEMATICS, APPLIED | MULTIPLE SOLUTIONS | SCHRODINGER-POISSON SYSTEMS | NONTRIVIAL SOLUTIONS | GROUND-STATE SOLUTIONS | MATHEMATICS | ENERGY SOLUTIONS

35B38 | Variational methods | 35J60 | Hartree-type nonlinearity | Mathematics, general | Symmetric mountain pass theorem | Mathematics | Kirchhoff equations | Sign-changing potential | EXISTENCE | MATHEMATICS, APPLIED | MULTIPLE SOLUTIONS | SCHRODINGER-POISSON SYSTEMS | NONTRIVIAL SOLUTIONS | GROUND-STATE SOLUTIONS | MATHEMATICS | ENERGY SOLUTIONS

Journal Article

Discrete and Continuous Dynamical Systems - Series S, ISSN 1937-1632, 12/2016, Volume 9, Issue 6, pp. 1613 - 1628

In this paper, we first give a sharp variational characterization to the smallest positive constant C-VGN in the following Variant Gagliardo-Nirenberg...

Variant Gagliardo-Nirenberg interpolation inequality | Minimal action solution | Hartree type nonlinearity | Sharp variational characterization | EXISTENCE | MATHEMATICS, APPLIED | minimal action solution | STABILITY | CALCULUS | CONCENTRATION-COMPACTNESS PRINCIPLE | STANDING WAVES | sharp variational characterization

Variant Gagliardo-Nirenberg interpolation inequality | Minimal action solution | Hartree type nonlinearity | Sharp variational characterization | EXISTENCE | MATHEMATICS, APPLIED | minimal action solution | STABILITY | CALCULUS | CONCENTRATION-COMPACTNESS PRINCIPLE | STANDING WAVES | sharp variational characterization

Journal Article

Monatshefte für Mathematik, ISSN 0026-9255, 2/2017, Volume 182, Issue 2, pp. 335 - 358

We consider the following singularly perturbed nonlocal elliptic problem $$\begin{aligned} -\left( \varepsilon ^{2}a+\varepsilon b\displaystyle \int _{\mathbb...

Kirchhoff-type equation | Ground state solution | 35B25 | 35J60 | Hartree-type nonlinearity | Mathematics, general | Mathematics | Variational method | 35B40 | Concentration phenomena | SCHRODINGER-EQUATIONS | MATHEMATICS | KIRCHHOFF-TYPE EQUATIONS | POSITIVE SOLUTIONS | CHOQUARD EQUATION | HARTREE-TYPE NONLINEARITIES

Kirchhoff-type equation | Ground state solution | 35B25 | 35J60 | Hartree-type nonlinearity | Mathematics, general | Mathematics | Variational method | 35B40 | Concentration phenomena | SCHRODINGER-EQUATIONS | MATHEMATICS | KIRCHHOFF-TYPE EQUATIONS | POSITIVE SOLUTIONS | CHOQUARD EQUATION | HARTREE-TYPE NONLINEARITIES

Journal Article

Results in Mathematics, ISSN 1422-6383, 3/2019, Volume 74, Issue 1, pp. 1 - 26

In this paper, we consider the following nonlinear problem of Kirchhoff-type with Hartree-type nonlinearities: $$\begin{aligned} \left\{ \begin{array}{ll}...

35J20 | positive solution | Pohožaev identity | 35J60 | ground state solution | Hartree-type nonlinearity | Mathematics, general | Mathematics | Kirchhoff equations | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | MULTIPLICITY | STATE SOLUTIONS | POSITIVE SOLUTIONS | HIGH-ENERGY SOLUTIONS | Pohoaev identity

35J20 | positive solution | Pohožaev identity | 35J60 | ground state solution | Hartree-type nonlinearity | Mathematics, general | Mathematics | Kirchhoff equations | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | MULTIPLICITY | STATE SOLUTIONS | POSITIVE SOLUTIONS | HIGH-ENERGY SOLUTIONS | Pohoaev identity

Journal Article

Applicable Analysis, ISSN 0003-6811, 12/2017, Volume 96, Issue 16, pp. 2846 - 2851

This paper deals with the Cauchy problem of the nonlinear Schrödinger equation with Combined Nonlinearities. By using the generalized Gagliardo-Nirenberg...

35Q55 | sharp criteria | global existence | Hartree-type | Nonlinear Schrödinger equation | blow-up solution | MATHEMATICS, APPLIED | Nonlinear Schrodinger equation | CAUCHY-PROBLEM | BLOW-UP | Formulas (mathematics) | Schroedinger equation | Cauchy problem

35Q55 | sharp criteria | global existence | Hartree-type | Nonlinear Schrödinger equation | blow-up solution | MATHEMATICS, APPLIED | Nonlinear Schrodinger equation | CAUCHY-PROBLEM | BLOW-UP | Formulas (mathematics) | Schroedinger equation | Cauchy problem

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 08/2018, Volume 464, Issue 1, pp. 402 - 420

In the present paper, we consider the following magnetic nonlinear Choquard equation{(−i∇+A(x))2u+μg(x)u=λu+(|x|−α⁎|u|2⁎α)|u|2α⁎−2uinRn,u∈H1(Rn,C) where n≥4,...

Nonlinear Schrödinger equations | Choquard equation | Magnetic potential | Hardy–Littlewood–Sobolev critical exponent | EXISTENCE | MATHEMATICS, APPLIED | NONLINEAR SCHRODINGER-EQUATIONS | SEMICLASSICAL SOLUTIONS | ELECTROMAGNETIC-FIELDS | LIMIT | GROUND-STATE SOLUTIONS | Nonlinear Schrodinger equations | UNIQUENESS | MATHEMATICS | BOUND-STATES | Hardy-Littlewood-Sobolev critical exponent | HARTREE-TYPE NONLINEARITIES | MULTIPLE POSITIVE SOLUTIONS

Nonlinear Schrödinger equations | Choquard equation | Magnetic potential | Hardy–Littlewood–Sobolev critical exponent | EXISTENCE | MATHEMATICS, APPLIED | NONLINEAR SCHRODINGER-EQUATIONS | SEMICLASSICAL SOLUTIONS | ELECTROMAGNETIC-FIELDS | LIMIT | GROUND-STATE SOLUTIONS | Nonlinear Schrodinger equations | UNIQUENESS | MATHEMATICS | BOUND-STATES | Hardy-Littlewood-Sobolev critical exponent | HARTREE-TYPE NONLINEARITIES | MULTIPLE POSITIVE SOLUTIONS

Journal Article

Lecture Notes in Engineering and Computer Science, ISSN 2078-0958, 2015, Volume 2, pp. 811 - 814

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