scholarly journals Classification of nonnegative solutions to an equation involving the Laplacian of arbitrary order

2019 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Phuong Le ◽  
Keyword(s):  
Arbitrary Order ◽  
Nonnegative Solutions

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 < 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 < γ < 1. As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities.

scholarly journals Classification of nonnegative solutions to static Schrödinger–Hartree–Maxwell system involving the fractional Laplacian

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yunting Li ◽  
Yaqiong Liu ◽  
Yunhui Yi
Keyword(s):  
Linear System ◽  
Fractional Laplacian ◽  
Critical Case ◽  
Direct Method ◽  
Maxwell System ◽  
Nonlocal Nonlinearity ◽  
Nonnegative Solutions ◽  
Moving Spheres ◽  
Funct Anal

AbstractThis paper is mainly concerned with the following semi-linear system involving the fractional Laplacian: $$ \textstyle\begin{cases} (-\Delta )^{\frac{\alpha }{2}}u(x)= (\frac{1}{ \vert \cdot \vert ^{\sigma }} \ast v^{p_{1}} )v^{p_{2}}(x), \quad x\in \mathbb{R}^{n}, \\ (-\Delta )^{\frac{\alpha }{2}}v(x)= (\frac{1}{ \vert \cdot \vert ^{\sigma }} \ast u^{q_{1}} )u^{q_{2}}(x), \quad x\in \mathbb{R}^{n}, \\ u(x)\geq 0,\quad\quad v(x)\geq 0, \quad x\in \mathbb{R}^{n}, \end{cases} $$ { ( − Δ ) α 2 u ( x ) = ( 1 | ⋅ | σ ∗ v p 1 ) v p 2 ( x ) , x ∈ R n , ( − Δ ) α 2 v ( x ) = ( 1 | ⋅ | σ ∗ u q 1 ) u q 2 ( x ) , x ∈ R n , u ( x ) ≥ 0 , v ( x ) ≥ 0 , x ∈ R n , where $0<\alpha \leq 2$ 0 < α ≤ 2 , $n\geq 2$ n ≥ 2 , $0<\sigma <n$ 0 < σ < n , and $0< p_{1}, q_{1}\leq \frac{2n-\sigma }{n-\alpha }$ 0 < p 1 , q 1 ≤ 2 n − σ n − α , $0< p_{2}, q_{2}\leq \frac{n+\alpha -\sigma }{n-\alpha }$ 0 < p 2 , q 2 ≤ n + α − σ n − α . Applying a variant (for nonlocal nonlinearity) of the direct method of moving spheres for fractional Laplacians, which was developed by W. Chen, Y. Li, and R. Zhang (J. Funct. Anal. 272(10):4131–4157, 2017), we derive the explicit forms for positive solution $(u,v)$ ( u , v ) in the critical case and nonexistence of positive solutions in the subcritical cases.

scholarly journals SINGULAR SOLUTIONS TO THE HEAT EQUATIONS WITH NONLINEAR ABSORPTION AND HARDY POTENTIALS

2012 ◽  
Vol 14 (02) ◽  
pp. 1250013 ◽  
Author(s):  
VITALI LISKEVICH ◽  
ANDREY SHISHKOV ◽  
ZEEV SOBOL
Keyword(s):  
Nonlinear Absorption ◽  
Singular Solution ◽  
Complete Classification ◽  
Beltrami Operator ◽  
Singular Solutions ◽  
Heat Equations ◽  
Nonnegative Solutions ◽  
The One ◽  
Very Singular Solution

We study the existence and nonexistence of singular solutions to the equation [Formula: see text], p > 1, in ℝN× [0, ∞), N ≥ 3, with a singularity at the point (0, 0), that is, nonnegative solutions satisfying u(x, 0) = 0 for x ≠ 0, assuming that α > -2 and [Formula: see text]. The problem is transferred to the one for a weighted Laplace–Beltrami operator with a nonlinear absorption, absorbing the Hardy potential in the weight. A classification of a singular solution to the weighted problem either as a source solution with a multiple of the Dirac mass as initial datum, or as a unique very singular solution, leads to a complete classification of singular solutions to the original problem, which exist if and only if [Formula: see text].

scholarly journals Classification of nonnegative solutions to fractional Schrödinger-Hatree-Maxwell type system

2021 ◽  
Vol 6 (12) ◽  
pp. 13665-13688
Author(s):  
Yaqiong Liu ◽  
◽  
Yunting Li ◽  
Qiuping Liao ◽  
Yunhui Yi
Keyword(s):  
Nonnegative Solution ◽  
Type System ◽  
Maximum Principles ◽  
Nonlocal Nonlinearity ◽  
Narrow Region ◽  
Nonnegative Solutions ◽  
Moving Spheres

<abstract><p>In this paper, we are concerned with the fractional Schrödinger-Hatree-Maxwell type system. We derive the forms of the nonnegative solution and classify nonlinearities by appling a variant (for nonlocal nonlinearity) of the direct moving spheres method for fractional Laplacians. The main ingredients are the variants (for nonlocal nonlinearity) of the maximum principles, i.e., <italic>narrow region principle</italic> (Theorem 2.3).</p></abstract>

scholarly journals Solution of one hypersingular integro-differential equation defined by determinants

2021 ◽  
pp. 17-28
Author(s):  
Andrei P. Shilin
Keyword(s):  
Differential Equation ◽  
Analytic Functions ◽  
Arbitrary Order ◽  
Exact Analytical Solution ◽  
Linear Equations ◽  
Linear Conjugation ◽  
Traditional Classification ◽  
Linear Differential ◽  
Problem Of Linear Conjugation

The paper provides an exact analytical solution to a hypersingular inregro-differential equation of arbitrary order. The equation is defined on a closed curve in the complex plane. A characteristic feature of the equation is that if is written using determinants. From the view of the traditional classification of the equations, it should be classified as linear equations with vatiable coefficients of a special form. The method of analytical continuation id applied. The equation is reduced to a boundary value problem of linear conjugation for analytic functions with some additional conditions. If this problem is solvable, if is required to solve two more linear differential equations in the class of analytic functions. The conditions of solvability are indicated explicitly. When these conditions are met, the solution can also be written explicitly. An example is given.

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