scholarly journals Nonradial Solutions for the Hénon Equation Close to the Threshold

2019 ◽  
Vol 19 (4) ◽  
pp. 757-770 ◽  
Author(s):  
Pablo Figueroa ◽  
Sérgio L. N. Neves
Keyword(s):  
Positive Integer ◽  
Unit Ball ◽  
Hénon Equation

AbstractWe consider the Hénon problem\left\{\begin{aligned} &\displaystyle{-}\Delta u=\lvert x\rvert^{\alpha}u^{% \frac{N+2+2\alpha}{N-2}-\varepsilon}&&\displaystyle\phantom{}\text{in }B_{1},% \\ &\displaystyle u>0&&\displaystyle\phantom{}\text{in }B_{1},\\ &\displaystyle u=0&&\displaystyle\phantom{}\text{on }\partial B_{1},\end{% aligned}\right.where {B_{1}} is the unit ball in {\mathbb{R}^{N}} and {N\geqslant 3}. For {\varepsilon>0} small enough, we use α as a parameter and prove the existence of a branch of nonradial solutions that bifurcates from the radial one when α is close to an even positive integer.

Least Energy Solutions and Group Invariant Solutions of the Hénon Equation

2013 ◽  
Vol 13 (3) ◽  
Author(s):  
Ryuji Kajikiya
Keyword(s):  
Unit Ball ◽  
Radial Symmetry ◽  
Coefficient Function ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Least Energy Solution ◽  
Hénon Equation ◽  
Group Invariant Solutions ◽  
Least Energy Solutions ◽  
Least Energy

AbstractIn this paper we study the generalized Hénon equation in the unit ball, where the coefficient function may or may not change its sign. We prove that the least energy solution is not radial and moreover we show the existence of a group invariant positive solution without radial symmetry.

scholarly journals Positive bounded solutions for nonlinear polyharmonic problems in the unit ball

2017 ◽  
Vol 25 (3) ◽  
pp. 143-153
Author(s):  
Habib Mâagli ◽  
Zagharide Zine El Abidine
Keyword(s):  
Fixed Point ◽  
Positive Integer ◽  
Fixed Point Theorem ◽  
Boundary Conditions ◽  
Unit Ball ◽  
Positive Solutions ◽  
Point Theorem ◽  
Bounded Solutions ◽  
Polyharmonic Equation ◽  
Existence Of Positive Solutions

Abstract In this paper, we study the existence of positive solutions for the following nonlinear polyharmonic equation (-∆)mu+λf(x, u) = 0 in B; subject to some boundary conditions, where m is a positive integer, λ is a nonnegative constant and B is the unit ball of ℝn (n ≥ 2). Under some appropriate assumptions on the nonnegative nonlinearity term f(x, u) and by using the Schäuder fixed point theorem, the existence of positive solutions is obtained. At last, examples are given for illustration.

scholarly journals The Lavrentiev gap phenomenon for harmonic maps into spheres holds on a dense set of zero degree boundary data

2017 ◽  
Vol 10 (3) ◽  
pp. 303-314 ◽  
Author(s):  
Katarzyna Mazowiecka ◽  
Paweł Strzelecki
Keyword(s):  
Positive Integer ◽  
Unit Ball ◽  
Boundary Data ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Singular Points ◽  
Dirichlet Energy ◽  
Zero Degree ◽  
Dense Set

AbstractWe consider the set of smooth zero degree maps {\psi\colon\mathbb{S}^{2}\to\mathbb{S}^{2}} which have the following properties: (i) There is a unique minimizing harmonic map {u\colon\mathbb{B}^{3}\to\mathbb{S}^{2}} which agrees with ψ on the boundary of the unit ball. (ii) The map u has at least N singular points in {\mathbb{B}^{3}}. (iii) The Lavrentiev gap phenomenon holds for ψ, i.e., the infimum of the Dirichlet energies {E(w)} of all smooth extensions {w\colon\mathbb{B}^{3}\to\mathbb{S}^{2}} of ψ is strictly larger than the Dirichlet energy {\int_{\mathbb{B}^{3}}\lvert\nabla u|^{2}} of the (irregular) minimizer u. For each positive integer N, we prove that this set is dense in the set {\mathcal{S}} of all smooth zero degree maps {\phi\colon\mathbb{S}^{2}\to\mathbb{S}^{2}} endowed with the {W^{1,p}}-topology, where 1 \leqp ¡ 2. This result is sharp since it fails in the {W^{1,2}}-topology of {\mathcal{S}}.

scholarly journals THE ROBIN PROBLEM FOR THE HÉNON EQUATION

2013 ◽  
Vol 88 (1) ◽  
pp. 1-11
Author(s):  
HAIYANG HE
Keyword(s):  
Symmetry Breaking ◽  
Unit Ball ◽  
Ground State ◽  
Existence Results ◽  
Robin Problem ◽  
Ground State Solutions ◽  
Hénon Equation ◽  
Image Position

AbstractIn this paper, we consider the following Robin problem:$$\begin{eqnarray*}\displaystyle \left\{ \begin{array}{ @{}ll@{}} \displaystyle - \Delta u= \mid x{\mathop{\mid }\nolimits }^{\alpha } {u}^{p} , \quad & \displaystyle x\in \Omega , \\ \displaystyle u\gt 0, \quad & \displaystyle x\in \Omega , \\ \displaystyle \displaystyle \frac{\partial u}{\partial \nu } + \beta u= 0, \quad & \displaystyle x\in \partial \Omega , \end{array} \right.&&\displaystyle\end{eqnarray*}$$where$\Omega $is the unit ball in${ \mathbb{R} }^{N} $centred at the origin, with$N\geq 3$,$p\gt 1$,$\alpha \gt 0$,$\beta \gt 0$, and$\nu $is the unit outward vector normal to$\partial \Omega $. We prove that the above problem has no solution when$\beta $is small enough. We also obtain existence results and we analyse the symmetry breaking of the ground state solutions.

scholarly journals Free abelian topological groups on countable CW-complexes

1990 ◽  
Vol 41 (3) ◽  
pp. 451-456 ◽  
Author(s):  
Eli Katz ◽  
Sidney A. Morris
Keyword(s):  
Topological Group ◽  
Positive Integer ◽  
Unit Ball ◽  
Closed Subgroup ◽  
Differentiable Manifold ◽  
Closed Unit Ball ◽  
Topological Groups ◽  
Abelian Topological Group ◽  
Cw Complex ◽  
Free Abelian Topological Group

Let n be a positive integer, Bn the closed unit ball in Euclidean n-space, and X any countable CW-complex of dimension at most n. It is shown that the free Abelian topological group on Bn, F(Bn), has F(X) as a closed subgroup. It is also shown that for every differentiable manifold Y of dimension at most n, F(Y) is a closed subgroup of F(Bn).

scholarly journals Singular solutions of a Hénon equation involving a nonlinear gradient term

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Craig Cowan ◽  
Abdolrahman Razani
Keyword(s):  
Unit Ball ◽  
Explicit Solution ◽  
Singular Solutions ◽  
Gradient Term ◽  
Hénon Equation ◽  
Smooth Perturbation

<p style='text-indent:20px;'>Here, we consider positive singular solutions of</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{array}{lcc} -\Delta u = |x|^\alpha |\nabla u|^p &amp; \text{in}&amp; \Omega \backslash\{0\},\\ u = 0&amp;\text{on}&amp; \partial \Omega, \end{array} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a small smooth perturbation of the unit ball in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ p $\end{document}</tex-math></inline-formula> are parameters in a certain range. Using an explicit solution on <inline-formula><tex-math id="M5">\begin{document}$ B_1 $\end{document}</tex-math></inline-formula> and a linearization argument, we obtain positive singular solutions on perturbations of the unit ball.</p>

Infinitely many non-radial solutions for the polyharmonic Hénon equation with a critical exponent

2017 ◽  
Vol 147 (2) ◽  
pp. 371-396 ◽  
Author(s):  
Yuxia Guo ◽  
Bo Li ◽  
Yi Li
Keyword(s):  
Critical Exponent ◽  
Unit Ball ◽  
Positive Solutions ◽  
Bounded Function ◽  
Radial Solutions ◽  
Hénon Equation ◽  
Image Position

We study the following polyharmonic Hénon equation:where (m)* = 2N/(N – 2m) is the critical exponent, B1(0) is the unit ball in ℝN, N ⩾ 2m + 2 and K(|y|) is a bounded function. We prove the existence of infinitely many non-radial positive solutions, whose energy can be made arbitrarily large.

The Order of Monochromatic Subgraphs with a Given Minimum Degree

10.37236/1725 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Yair Caro ◽  
Raphael Yuster
Keyword(s):  
Positive Integer ◽  
Minimum Degree

Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the largest integer $t$ such that in every coloring of the edges of $G$ with two colors there is a monochromatic subgraph with minimum degree at least $d$ and order at least $t$. Let $f_G(d)=0$ in case there is a $2$-coloring of the edges of $G$ with no such monochromatic subgraph. Let $f(n,k,d)$ denote the minimum of $f_G(d)$ where $G$ ranges over all graphs with $n$ vertices and minimum degree at least $k$. In this paper we establish $f(n,k,d)$ whenever $k$ or $n-k$ are fixed, and $n$ is sufficiently large. We also consider the case where more than two colors are allowed.

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