scholarly journals On critical exponents of a k-Hessian equation in the whole space

2019 ◽  
Vol 149 (6) ◽  
pp. 1555-1575 ◽  
Author(s):  
Yun Wang ◽  
Yutian Lei
Keyword(s):  
Critical Exponents ◽  
Decay Rates ◽  
Classical Solutions ◽  
Sobolev Type ◽  
Extremal Functions ◽  
Hessian Equation ◽  
Hessian Equations ◽  
Radial Structure ◽  
Whole Space ◽  
Image Position

AbstractIn this paper, we study negative classical solutions and stable solutions of the following k-Hessian equation $$F_k(D^2V) = (-V)^p\quad {\rm in}\;\; R^n$$with radial structure, where n ⩾ 3, 1 < k < n/2 and p > 1. This equation is related to the extremal functions of the Hessian Sobolev inequality on the whole space. Several critical exponents including the Serrin type, the Sobolev type, and the Joseph-Lundgren type, play key roles in studying existence and decay rates. We believe that these critical exponents still come into play to research k-Hessian equations without radial structure.

Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity

2018 ◽  
Vol 149 (04) ◽  
pp. 979-994 ◽  
Author(s):  
Daomin Cao ◽  
Wei Dai
Keyword(s):  
Critical Case ◽  
Classical Solutions ◽  
Sobolev Inequalities ◽  
Extremal Functions ◽  
Method Of Moving Planes ◽  
Nonnegative Solutions ◽  
Moving Planes ◽  
Image Position ◽  
Harmonic Equation

AbstractIn this paper, we are concerned with the following bi-harmonic equation with Hartree type nonlinearity $$\Delta ^2u = \left( {\displaystyle{1 \over { \vert x \vert ^8}}* \vert u \vert ^2} \right)u^\gamma ,\quad x\in {\open R}^d,$$where 0 &lt; γ ⩽ 1 and d ⩾ 9. By applying the method of moving planes, we prove that nonnegative classical solutions u to (𝒫γ) are radially symmetric about some point x0 ∈ ℝd and derive the explicit form for u in the Ḣ2 critical case γ = 1. We also prove the non-existence of nontrivial nonnegative classical solutions in the subcritical cases 0 &lt; γ &lt; 1. As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities.

A Quasi-Linear Elliptic Boundary Value Problem

1966 ◽  
Vol 18 ◽  
pp. 1105-1112 ◽  
Author(s):  
R. A. Adams
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Classical Solutions ◽  
Elliptic Boundary Value Problem ◽  
Elliptic Boundary Value ◽  
Open Set ◽  
Image Position ◽  
Linear Elliptic Boundary ◽  
Typical Point

Let Ω be a bounded open set in Euclidean n-space, En. Let α = (α1, … , an) be an n-tuple of non-negative integers;and denote by Qm the set ﹛α| 0 ⩽ |α| ⩽ m}. Denote by x = (x1, … , xn) a typical point in En and putIn this paper we establish, under certain circumstances, the existence of weak and classical solutions of the quasi-linear Dirichlet problem1

scholarly journals ON LAGUERRE–SOBOLEV TYPE ORTHOGONAL POLYNOMIALS: ZEROS AND ELECTROSTATIC INTERPRETATION

2013 ◽  
Vol 55 (1) ◽  
pp. 39-54
Author(s):  
LUIS ALEJANDRO MOLANO MOLANO
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Inner Product ◽  
Sobolev Type ◽  
Electrostatic Model ◽  
Order Linear Differential Equation ◽  
Standard Methods ◽  
Linear Differential ◽  
Electrostatic Interpretation ◽  
Image Position

AbstractWe study the sequence of monic polynomials orthogonal with respect to inner product $$\begin{eqnarray*}\langle p, q\rangle = \int \nolimits \nolimits_{0}^{\infty } p(x)q(x){e}^{- x} {x}^{\alpha } \hspace{0.167em} dx+ Mp(\zeta )q(\zeta )+ N{p}^{\prime } (\zeta ){q}^{\prime } (\zeta ),\end{eqnarray*}$$ where $\alpha \gt - 1$, $M\geq 0$, $N\geq 0$, $\zeta \lt 0$, and $p$ and $q$ are polynomials with real coefficients. We deduce some interlacing properties of their zeros and, by using standard methods, we find a second-order linear differential equation satisfied by the polynomials and discuss an electrostatic model of their zeros.

scholarly journals Decay estimates for solutions of nonlocal semilinear equations

2015 ◽  
Vol 218 ◽  
pp. 175-198 ◽  
Author(s):  
Marco Cappiello ◽  
Todor Gramchev ◽  
Luigi Rodino
Keyword(s):  
Water Waves ◽  
Fourier Multiplier ◽  
Sobolev Type ◽  
Decay Estimates ◽  
Shallow Water Waves ◽  
Algebraic Decay ◽  
Semilinear Equations ◽  
Nonlocal Equations ◽  
Image Position ◽  
Strong Contrast

AbstractWe investigate the decay for |x|→∞ of weak Sobolev-type solutions of semilinear nonlocal equations Pu = F(u). We consider the case when P = p(D) is an elliptic Fourier multiplier with polyhomogeneous symbol p(ξ), and we derive algebraic decay estimates in terms of weighted Sobolev norms. Our basic example is the celebrated Benjamin–Ono equation for internal solitary waves of deep stratified fluids. Their profile presents algebraic decay, in strong contrast with the exponential decay for KdV shallow water waves.

Singular limit of a dispersive Navier–Stokes system with an entropy structure

2014 ◽  
Vol 13 (01) ◽  
pp. 77-99
Author(s):  
C. David Levermore ◽  
Weiran Sun ◽  
Konstantina Trivisa
Keyword(s):  
Stokes System ◽  
Singular Limit ◽  
Navier Stokes ◽  
Classical Solutions ◽  
Third Order ◽  
Fluid System ◽  
Low Mach Number ◽  
Low Mach Number Limit ◽  
Whole Space ◽  
Entropy Structure

We prove a low Mach number limit for a dispersive fluid system [3] which contains third-order corrections to the compressible Navier–Stokes. We show that the classical solutions to this system in the whole space ℝn converge to classical solutions to ghost-effect systems [7]. Our analysis follows the framework in [4], which is built on the methodology developed by Métivier and Schochet [6] and Alazard [1] for systems up to the second order. The key new ingredient is the application of the entropy structure of the dispersive fluid system. This structure enables us to treat cases not covered in [4] and to simplify the analysis in [4].

L∞C(ℝn)-decay of classical solutions for nonlinear Schrödinger equations

1986 ◽  
Vol 104 (3-4) ◽  
pp. 309-327 ◽  
Author(s):  
Nakao Hayashi ◽  
Masayoshi Tsutsumi
Keyword(s):  
Schrödinger Equation ◽  
Initial Value Problem ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Classical Solutions ◽  
Initial Value ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Sign Conditions ◽  
Image Position

SynopsisWe study the initial value problem for the nonlinear Schrödinger equationUnder suitable regularity assumptions on f and ø and growth and sign conditions on f, it is shown that the maximum norms of solutions to (*) decay as t→² ∞ at the same rate as that of solutions to the free Schrödinger equation.

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