Mathematische annalen, ISSN 1432-1807, 2017, Volume 370, Issue 1-2, pp. 447 - 489

We extend and improve the results in Dong and Kim (Commun Partial Differ Equ 42(3):417–435, 2017): showing that weak solutions to full elliptic equations in...

35B65 | Mathematics, general | Mathematics | Primary 35B45 | Secondary 35J47

35B65 | Mathematics, general | Mathematics | Primary 35B45 | Secondary 35J47

Journal Article

Nonlinear Differential Equations and Applications NoDEA, ISSN 1021-9722, 12/2015, Volume 22, Issue 6, pp. 1897 - 1910

Here, we investigate the existence of solutions to a stationary mean-field game model introduced by J.-M. Lasry and P.-L. Lions. This model features a...

Mean-field games | Analysis | 35A01 | Quadratic Hamiltonians | Mathematics | Congestion | 35J47 | MATHEMATICS, APPLIED

Mean-field games | Analysis | 35A01 | Quadratic Hamiltonians | Mathematics | Congestion | 35J47 | MATHEMATICS, APPLIED

Journal Article

Advances in nonlinear analysis, ISSN 2191-950X, 2018, Volume 7, Issue 4, pp. 425 - 447

The paper is concerned with a priori estimates of positive solutions of quasilinear elliptic systems of equations or inequalities in an open set of associated...

positive solutions | Quasilinear elliptic systems | Liouville theorems | 35B45 | 35J92 | 35B40 | 35J47 | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | EQUATIONS

positive solutions | Quasilinear elliptic systems | Liouville theorems | 35B45 | 35J92 | 35B40 | 35J47 | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | EQUATIONS

Journal Article

Communications in Partial Differential Equations, ISSN 0360-5302, 04/2020, Volume 45, Issue 4, pp. 285 - 302

We establish the existence of finitely many sign-changing solutions to the Lane-Emden system where the exponents p and q lie on the critical hyperbola These...

variational methods | critical hyperbola | Concentration-compactness | symmetries | MSC2010: 35J47 | entire nodal solutions | Hamiltonian system | MATHEMATICS | MATHEMATICS, APPLIED | MULTIPLE SOLUTIONS | SYMMETRY | ELLIPTIC-SYSTEMS | EQUATION

variational methods | critical hyperbola | Concentration-compactness | symmetries | MSC2010: 35J47 | entire nodal solutions | Hamiltonian system | MATHEMATICS | MATHEMATICS, APPLIED | MULTIPLE SOLUTIONS | SYMMETRY | ELLIPTIC-SYSTEMS | EQUATION

Journal Article

5.
Full Text
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

Archive for Rational Mechanics and Analysis, ISSN 0003-9527, 11/2015, Volume 218, Issue 2, pp. 647 - 697

For a class of systems of semi-linear elliptic equations, including for p = 2 (variational-type interaction) or p = 1 (symmetric-type interaction), we prove...

MATHEMATICS, APPLIED | MODELING PHASE-SEPARATION | MECHANICS | FREE-BOUNDARIES | SPATIAL SEGREGATION | THEOREMS | EQUATIONS | MONOTONICITY | BOSE-EINSTEIN CONDENSATION | ELLIPTIC SYSTEM | DIFFUSION-SYSTEMS | CONJECTURE | Mathematics - Analysis of PDEs

MATHEMATICS, APPLIED | MODELING PHASE-SEPARATION | MECHANICS | FREE-BOUNDARIES | SPATIAL SEGREGATION | THEOREMS | EQUATIONS | MONOTONICITY | BOSE-EINSTEIN CONDENSATION | ELLIPTIC SYSTEM | DIFFUSION-SYSTEMS | CONJECTURE | Mathematics - Analysis of PDEs

Journal Article

Advances in Nonlinear Analysis, ISSN 2191-9496, 05/2018, Volume 7, Issue 2, pp. 139 - 148

In the very recent paper [ ], the second author proved that for any , the fully nonlinear first order system is well posed in the so-called J. L. Lions space...

Baire category method | 35J60 | elliptic first order systems | Cauchy–Riemann equations | 35J47 | 35J46 | compensated compactness | convex integration | 32W50 | calculus of variations | Cordes’ condition | 35D30 | fully nonlinear systems | 32A50 | Campanato’s near operators | Campanato's near operators | Cordes' condition | Cauchy-Riemann equations | MATHEMATICS | MATHEMATICS, APPLIED

Baire category method | 35J60 | elliptic first order systems | Cauchy–Riemann equations | 35J47 | 35J46 | compensated compactness | convex integration | 32W50 | calculus of variations | Cordes’ condition | 35D30 | fully nonlinear systems | 32A50 | Campanato’s near operators | Campanato's near operators | Cordes' condition | Cauchy-Riemann equations | MATHEMATICS | MATHEMATICS, APPLIED

Journal Article

Annali di Matematica Pura ed Applicata, ISSN 0373-3114, 6/2014, Volume 193, Issue 3, pp. 837 - 858

Under the assumption that the coefficients are regularly varying functions, existence and asymptotic form of strongly decreasing solutions are here studied for...

Quasilinear system | 34C11 | 35J92 | Emden–Fowler system | 34C41 | Mathematics, general | Mathematics | Lane–Emden system | Decreasing solution | Regular variation | 35J47 | 34E05 | Lane-Emden system | Emden-Fowler system | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | POSITIVE SOLUTIONS | BEHAVIOR | DIFFERENTIAL-EQUATIONS

Quasilinear system | 34C11 | 35J92 | Emden–Fowler system | 34C41 | Mathematics, general | Mathematics | Lane–Emden system | Decreasing solution | Regular variation | 35J47 | 34E05 | Lane-Emden system | Emden-Fowler system | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | POSITIVE SOLUTIONS | BEHAVIOR | DIFFERENTIAL-EQUATIONS

Journal Article

Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, 12/2016, Volume 55, Issue 6, pp. 1 - 58

We study the nonlocal Schrödinger–Poisson–Slater type equation $$\begin{aligned} - \Delta u + (I_\alpha *\vert u\vert ^p)\vert u\vert ^{p - 2} u= \vert u\vert...

Mathematics | Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | 35Q55 (35J91, 35J47, 35J50, 31B35)

Mathematics | Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | 35Q55 (35J91, 35J47, 35J50, 31B35)

Journal Article

Journal of Dynamics and Differential Equations, ISSN 1040-7294, 12/2018, Volume 30, Issue 4, pp. 1365 - 1388

Mean-field games (MFGs) are models of large populations of rational agents who seek to optimize an objective function that takes into account their location...

Mean-field games | Ordinary Differential Equations | 35A01 | Mathematics | Stationary problems | Applications of Mathematics | Congestion problems | 35J47 | Partial Differential Equations | MATHEMATICS | MATHEMATICS, APPLIED

Mean-field games | Ordinary Differential Equations | 35A01 | Mathematics | Stationary problems | Applications of Mathematics | Congestion problems | 35J47 | Partial Differential Equations | MATHEMATICS | MATHEMATICS, APPLIED

Journal Article

Advances in Nonlinear Analysis, ISSN 2191-9496, 05/2017, Volume 6, Issue 2, pp. 165 - 182

In this paper we give a classification of positive radial solutions of the following system: in the open ball , with , and , , nonnegative nondecreasing...

Elliptic systems | explosive solutions | coercive systems | 35J47 | 35B44 | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | INEQUALITIES | EQUATIONS

Elliptic systems | explosive solutions | coercive systems | 35J47 | 35B44 | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | INEQUALITIES | EQUATIONS

Journal Article

Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, 2/2018, Volume 57, Issue 1, pp. 1 - 20

We establish the existence of a positive fully nontrivial solution (u, v) to the weakly coupled elliptic system $$\begin{aligned} {\left\{ \begin{array}{ll}...

35J20 | Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | 35B33 | Mathematics | 35B40 | 35J47 | 35B08 | MATHEMATICS | MATHEMATICS, APPLIED | R-N | WAVES | NONLINEAR SCHRODINGER-EQUATIONS | CRITICAL EXPONENT | BOUND-STATES | POSITIVE SOLUTIONS

35J20 | Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | 35B33 | Mathematics | 35B40 | 35J47 | 35B08 | MATHEMATICS | MATHEMATICS, APPLIED | R-N | WAVES | NONLINEAR SCHRODINGER-EQUATIONS | CRITICAL EXPONENT | BOUND-STATES | POSITIVE SOLUTIONS

Journal Article

Journal of Applied Analysis, ISSN 1425-6908, 12/2017, Volume 23, Issue 2, pp. 137 - 140

We briefly discuss the notion of the Lagrange multiplier for a linear constraint in the Hilbert space setting, and we prove that the pressure appearing in the...

pressure function | Lagrange multiplier | 49K20 | 76A02 | 35Q35 | 76B03 | 49J50 | Banach closed range theorem | 35J47 | Stokes equations

pressure function | Lagrange multiplier | 49K20 | 76A02 | 35Q35 | 76B03 | 49J50 | Banach closed range theorem | 35J47 | Stokes equations

Journal Article

Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, 10/2017, Volume 56, Issue 5, pp. 1 - 13

For the singularly perturbed system $$\begin{aligned} \varDelta u_{i,\beta }=\beta u_{i,\beta }\sum _{j\ne i}u_{j,\beta }^2, \quad 1\le i\le N, \end{aligned}$$...

35B25 | Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | Mathematics | 35J47 | PHASE-SEPARATION | MATHEMATICS | EQUATIONS | MATHEMATICS, APPLIED | SYSTEMS | CONJECTURE

35B25 | Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | Mathematics | 35J47 | PHASE-SEPARATION | MATHEMATICS | EQUATIONS | MATHEMATICS, APPLIED | SYSTEMS | CONJECTURE

Journal Article

Communications in Partial Differential Equations, ISSN 0360-5302, 08/2016, Volume 41, Issue 8, pp. 1324 - 1346

We consider stationary viscous mean-field games (MFG) systems in the case of local, decreasing and unbounded coupling. These systems arise in ergodic MFG...

Gagliardo-Nirenberg inequality | Concentration | Pohozaev identity | critical exponent | Gagliardo–Nirenberg inequality | EXISTENCE | MATHEMATICS, APPLIED | POSITIVE SOLUTIONS | 35B33 | 35J47 | ORBITAL STABILITY | UNIQUENESS | MATHEMATICS | COST | LONG-TIME AVERAGE | 49N70 | NONLINEAR ELLIPTIC-EQUATIONS | MASS | GROUND-STATES | Partial differential equations | Economic models | Brownian movements | Infinity | Uniqueness | Games | Agglomeration | Coupling | Calibration | Regularity | Invariants

Gagliardo-Nirenberg inequality | Concentration | Pohozaev identity | critical exponent | Gagliardo–Nirenberg inequality | EXISTENCE | MATHEMATICS, APPLIED | POSITIVE SOLUTIONS | 35B33 | 35J47 | ORBITAL STABILITY | UNIQUENESS | MATHEMATICS | COST | LONG-TIME AVERAGE | 49N70 | NONLINEAR ELLIPTIC-EQUATIONS | MASS | GROUND-STATES | Partial differential equations | Economic models | Brownian movements | Infinity | Uniqueness | Games | Agglomeration | Coupling | Calibration | Regularity | Invariants

Journal Article

Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, 4/2018, Volume 57, Issue 2, pp. 1 - 26

In this paper we establish existence of radial and nonradial solutions to the system $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u_1 =...

Primary 35J47 | Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | 35B33 | 35B32 | Secondary 35B09 | Mathematics | 35B08 | MATHEMATICS | MATHEMATICS, APPLIED | ELLIPTIC SYSTEM

Primary 35J47 | Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | 35B33 | 35B32 | Secondary 35B09 | Mathematics | 35B08 | MATHEMATICS | MATHEMATICS, APPLIED | ELLIPTIC SYSTEM

Journal Article

Advances in Nonlinear Analysis, ISSN 2191-9496, 05/2017, Volume 6, Issue 2, pp. 99 - 120

The core of this paper concerns the existence (via regularity) of weak solutions in of a class of elliptic systems such as deriving from saddle points of...

49J35 | Elliptic systems | 35J47 | 35J25 | saddle points | calculus of variations | MATHEMATICS | MATHEMATICS, APPLIED

49J35 | Elliptic systems | 35J47 | 35J25 | saddle points | calculus of variations | MATHEMATICS | MATHEMATICS, APPLIED

Journal Article

Annali di matematica pura ed applicata, ISSN 1618-1891, 2019, Volume 198, Issue 4, pp. 1243 - 1255

We study a weakly coupled supercritical elliptic system of the form $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = |x_2|^\gamma \left( \mu...

Bounded domain | 35J50 | Phase separation | 35B33 | Hénon-type equation | Mathematics, general | Mathematics | 35B40 | 35J47 | Supercritical nonlinearity | Weakly coupled elliptic system | SCHRODINGER-EQUATIONS | MATHEMATICS | MATHEMATICS, APPLIED | Henon-type equation | CRITICAL EXPONENT

Bounded domain | 35J50 | Phase separation | 35B33 | Hénon-type equation | Mathematics, general | Mathematics | 35B40 | 35J47 | Supercritical nonlinearity | Weakly coupled elliptic system | SCHRODINGER-EQUATIONS | MATHEMATICS | MATHEMATICS, APPLIED | Henon-type equation | CRITICAL EXPONENT

Journal Article

Mathematische Annalen, ISSN 0025-5831, 02/2018, Volume 370, Issue 1-2, pp. 447 - 489

Journal Article

Annali di Matematica Pura ed Applicata (1923 -), ISSN 0373-3114, 6/2019, Volume 198, Issue 3, pp. 821 - 836

Let $$\varOmega \subset \mathbb {R}^N$$ Ω ⊂ R N , $$N\ge 2$$ N ≥ 2 , be a bounded domain which is divided into two sub-domains $$\varOmega _1$$ Ω 1 and...

35J50 | Sobolev space | Neumann boundary condition | Mathematics | 35J47 | 35J57 | Robin boundary condition | Transmission problem | Lusternik–Schnirelmann principle | Lagrange multipliers | Eigenvalues | Mathematics, general | Riemannian setting | MATHEMATICS | MATHEMATICS, APPLIED | Lusternik-Schnirelmann principle

35J50 | Sobolev space | Neumann boundary condition | Mathematics | 35J47 | 35J57 | Robin boundary condition | Transmission problem | Lusternik–Schnirelmann principle | Lagrange multipliers | Eigenvalues | Mathematics, general | Riemannian setting | MATHEMATICS | MATHEMATICS, APPLIED | Lusternik-Schnirelmann principle

Journal Article

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