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 Sharp Sobolev Inequalities via Projection Averages

2020 ◽  
Author(s):  
Philipp Kniefacz ◽  
Franz E. Schuster
Keyword(s):  
Bounded Variation ◽  
Sobolev Inequality ◽  
Complete Classification ◽  
Functions Of Bounded Variation ◽  
Sobolev Inequalities ◽  
Extremal Functions ◽  
Affine Invariant ◽  
Entire Family ◽  
New Inequalities

Abstract A family of sharp $$L^p$$ L p Sobolev inequalities is established by averaging the length of i-dimensional projections of the gradient of a function. Moreover, it is shown that each of these new inequalities directly implies the classical $$L^p$$ L p  Sobolev inequality of Aubin and Talenti and that the strongest member of this family is the only affine invariant one among them—the affine $$L^p$$ L p  Sobolev inequality of Lutwak, Yang, and Zhang. When $$p = 1$$ p = 1 , the entire family of new Sobolev inequalities is extended to functions of bounded variation to also allow for a complete classification of all extremal functions in this case.

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.

scholarly journals Symmetry and Nonexistence of Positive Solutions for a Fractional Laplacion System with Coupled Terms

2022 ◽  
Author(s):  
Rong Zhang
Keyword(s):  
Positive Solution ◽  
Elliptic System ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Critical Case ◽  
Nonlinear Elliptic System ◽  
Subcritical Case ◽  
Nonlinear Elliptic ◽  
Method Of Moving Planes ◽  
Moving Planes

Abstract In this paper, we study the problem for a nonlinear elliptic system involving fractional Laplacion: (equation 1.1) where 0 < α, β < 2, p, q > 0 and max{p, q} ≥ 1, α + γ > 0, β + τ > 0, n ≥ 2. First of all, while in the subcritical case, i.e. n + α + γ − p(n − α) − (q + 1)(n − β) > 0, n + β + τ − (p + 1)(n − α) − q(n − β) > 0, we prove the nonexistence of positive solution for the above system in R n . Moreover, though Doubling Lemma to obtain the singularity estimates of the positive solution on bounded domain Ω. In addition, while in the critical case, i.e. n+α+γ −p(n−α)−(q + 1)(n−β) = 0, n+β +τ −(p+ 1)(n−α)−q(n−β) = 0, we show that the positive solution of above system are radical symmetric and decreasing about some point by using the method of Moving planes in Rn Mathematics Subject Classification (2020): 35R11, 35A10, 35B06.

Homogeneous sharp Sobolev inequalities on product manifolds

2006 ◽  
Vol 136 (2) ◽  
pp. 277-300 ◽  
Author(s):  
Jurandir Ceccon ◽  
Marcos Montenegro
Keyword(s):  
Riemannian Manifolds ◽  
Sobolev Inequality ◽  
Sobolev Inequalities ◽  
Extremal Functions ◽  
Mass Transportation ◽  
Best Constant ◽  
First Order ◽  
Image Position ◽  
Optimal Sobolev Inequality ◽  
Homogeneous Convex

Let (M, g) and (N, h) be compact Riemannian manifolds of dimensions m and n, respectively. For p-homogeneous convex functions f(s, t) on [0,∞) × [0, ∞), we study the validity and non-validity of the first-order optimal Sobolev inequality on H1, p(M × N) where and Kf = Kf (m, n, p) is the best constant of the homogeneous Sobolev inequality on D1, p (Rm+n), The proof of the non-validity relies on the knowledge of extremal functions associated with the Sobolev inequality above. In order to obtain such extremals we use mass transportation and convex analysis results. Since variational arguments do not work for general functions f, we investigate the validity in a uniform sense on f and argue with suitable approximations of f which are also essential in the non-validity. Homogeneous Sobolev inequalities on product manifolds are connected to elliptic problems involving a general class of operators.

scholarly journals Radial symmetry of nonnegative solutions for nonlinear integral systems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zhenjie Li ◽  
Chunqin Zhou
Keyword(s):  
Singular Integral ◽  
Nonnegative Solution ◽  
Radial Symmetry ◽  
Monotonicity Condition ◽  
Independent Variables ◽  
Method Of Moving Planes ◽  
Integral Forms ◽  
Nonnegative Solutions ◽  
Integral Systems ◽  
Moving Planes

<p style='text-indent:20px;'>In this paper, we investigate the nonnegative solutions of the nonlinear singular integral system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \left\{ \begin{array}{lll} u_i(x) = \int_{\mathbb{R}^n}\frac{1}{|x-y|^{n-\alpha}|y|^{a_i}}f_i(u(y))dy,\quad x\in\mathbb{R}^n,\quad i = 1,2\cdots,m,\\ 0&lt;\alpha&lt;n,\quad u(x) = (u_1(x),\cdots,u_m(x)),\nonumber \end{array}\right. \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ 0&lt;a_i/2&lt;\alpha $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ f_i(u) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 1\leq i\leq m $\end{document}</tex-math></inline-formula>, are real-valued functions, nonnegative and monotone nondecreasing with respect to the independent variables <inline-formula><tex-math id="M4">\begin{document}$ u_1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ u_2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ \cdots $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ u_m $\end{document}</tex-math></inline-formula>. By the method of moving planes in integral forms, we show that the nonnegative solution <inline-formula><tex-math id="M8">\begin{document}$ u = (u_1,u_2,\cdots,u_m) $\end{document}</tex-math></inline-formula> is radially symmetric when <inline-formula><tex-math id="M9">\begin{document}$ f_i $\end{document}</tex-math></inline-formula> satisfies some monotonicity condition.</p>

scholarly journals Radial Symmetry and Monotonicity of Solutions to a System Involving Fractional p-Laplacian in a Ball

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Linfen Cao ◽  
Xiaoshan Wang ◽  
Zhaohui Dai
Keyword(s):  
Nonlinear System ◽  
Unit Ball ◽  
Positive Solutions ◽  
Direct Method ◽  
Radial Symmetry ◽  
Continuous Functions ◽  
Method Of Moving Planes ◽  
Moving Planes

In this paper, we study a nonlinear system involving the fractional p-Laplacian in a unit ball and establish the radial symmetry and monotonicity of its positive solutions. By using the direct method of moving planes, we prove the following result. For 0<s,t<1,p>0, if u and v satisfy the following nonlinear system -Δpsux=fvx;  -Δptvx=gux,  x∈B10;  ux,vx=0,  x∉B10. and f,g are nonnegative continuous functions satisfying the following: (i) f(r) and g(r) are increasing for r>0; (ii) f′(r)/rp-2, g′(r)/rp-2 are bounded near r=0. Then the positive solutions (u,v) must be radially symmetric and monotone decreasing about the origin.

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

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