Fast and slow decay solutions for supercritical fractional elliptic problems in exterior domains

2021 ◽  
pp. 1-26
Author(s):  
Weiwei Ao ◽  
Chao Liu ◽  
Liping Wang
Keyword(s):  
Unit Ball ◽  
Elliptic Problem ◽  
Elliptic Problems ◽  
Infinitely Many Solutions ◽  
Fast Decay ◽  
Exterior Domains ◽  
Slow Decay ◽  
Image Position ◽  
Partial Differ

We consider the fractional elliptic problem: where B1 is the unit ball in ℝ N , N ⩾ 3, s ∈ (0, 1) and p > (N + 2s)/(N − 2s). We prove that this problem has infinitely many solutions with slow decay O(|x|−2s/(p−1)) at infinity. In addition, for each s ∈ (0, 1) there exists P s  > (N + 2s)/(N − 2s), for any (N + 2s)/(N − 2s) < p < P s , the above problem has a solution with fast decay O(|x|2s−N). This result is the extension of the work by Dávila, del Pino, Musso and Wei (2008, Calc. Var. Partial Differ. Equ. 32, no. 4, 453–480) to the fractional case.

Multiplicity of non-radial solutions of critical elliptic problems in an annulus

2005 ◽  
Vol 135 (1) ◽  
pp. 25-37 ◽  
Author(s):  
Djairo Guedes de Figueiredo ◽  
Olímpio Hiroshi Miyagaki
Keyword(s):  
Critical Points ◽  
Elliptic Problem ◽  
Elliptic Problems ◽  
Critical Sobolev Exponent ◽  
Radial Solutions ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Sobolev Exponent ◽  
Image Position

By looking for critical points of functionals defined in some subspaces of , invariant under some subgroups of O (N), we prove the existence of many positive non-radial solutions for the following semilinear elliptic problem involving critical Sobolev exponent on an annulus, where 2* − 1 := (N + 2)/(N − 2) (N ≥ 4), the domain is an annulus and f : R+ × R+ → R is a C1 function, which is a subcritical perturbation.

Infinitely Many Solutions for Quasilinear Elliptic Problems with Broken Symmetry

2013 ◽  
Vol 13 (3) ◽  
Author(s):  
Rossella Bartolo
Keyword(s):  
Bounded Domain ◽  
Elliptic Problem ◽  
Existence Of Solutions ◽  
Elliptic Problems ◽  
Broken Symmetry ◽  
Infinitely Many Solutions ◽  
Quasilinear Elliptic Problem ◽  
Quasilinear Elliptic Problems ◽  
Quasilinear Elliptic

AbstractThe aim of this paper is investigating the existence of solutions of the quasilinear elliptic Problemwhere Ω is an open bounded domain of R

scholarly journals EXISTENCE OF INFINITELY MANY SOLUTIONS FOR SUBLINEAR ELLIPTIC PROBLEMS

2012 ◽  
Vol 54 (3) ◽  
pp. 535-545
Author(s):  
X. ZHONG ◽  
W. ZOU
Keyword(s):  
Boundary Value Problem ◽  
Bounded Domain ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Boundary Value ◽  
Elliptic Problems ◽  
Infinitely Many Solutions ◽  
Dirichlet Boundary Value Problem ◽  
Image Position

AbstractWe study the following nonlinear Dirichlet boundary value problem: where Ω is a bounded domain in ℝN(N ≥ 2) with a smooth boundary ∂Ω and g ∈ C(Ω × ℝ) is a function satisfying $\displaystyle \underset{|t|\rightarrow 0}{\lim}\frac{g(x, t)}{t}= \infty$ for all x ∈ Ω. Under appropriate assumptions, we prove the existence of infinitely many solutions when g(x, t) is not odd in t.

Fast and slow decay solutions for supercritical elliptic problems in exterior domains

2008 ◽  
Vol 32 (4) ◽  
pp. 453-480 ◽  
Author(s):  
Juan Dávila ◽  
Manuel del Pino ◽  
Monica Musso ◽  
Juncheng Wei
Keyword(s):  
Elliptic Problems ◽  
Exterior Domains ◽  
Slow Decay

Kolmogorov, Linear and Pseudo-Dimensional Widths of Classes of s-Monotone Functions in 𝕃p, 0 < p < 1

2008 ◽  
Vol 51 (2) ◽  
pp. 236-248
Author(s):  
Victor N. Konovalov ◽  
Kirill A. Kopotun
Keyword(s):  
Unit Ball ◽  
Finite Interval ◽  
Divided Differences ◽  
Monotone Functions ◽  
Image Position

AbstractLet Bp be the unit ball in 𝕃p, 0 < p < 1, and let , s ∈ ℕ, be the set of all s-monotone functions on a finite interval I, i.e., consists of all functions x : I ⟼ ℝ such that the divided differences [x; t0, … , ts] of order s are nonnegative for all choices of (s + 1) distinct points t0, … , ts ∈ I. For the classes Bp := ∩ Bp, we obtain exact orders of Kolmogorov, linear and pseudo-dimensional widths in the spaces , 0 < q < p < 1:

Three-phase boundary motions under constant velocities. I: The vanishing surface tension limit

1996 ◽  
Vol 126 (4) ◽  
pp. 837-865 ◽  
Author(s):  
Fernando Reitich ◽  
H. Mete Soner
Keyword(s):  
Surface Tension ◽  
Formal Analysis ◽  
Local Equilibrium ◽  
Convergence Result ◽  
Normal Velocity ◽  
Infinitely Many Solutions ◽  
Weak Formulation ◽  
Material Interfaces ◽  
Internal Surfaces ◽  
Image Position

In this paper, we deal with the dynamics of material interfaces such as solid–liquid, grain or antiphase boundaries. We concentrate on the situation in which these internal surfaces separate three regions in the material with different physical attributes (e.g. grain boundaries in a polycrystal). The basic two-dimensional model proposes that the motion of an interface Гij between regions i and j (i, j = 1, 2, 3, i ≠ j) is governed by the equationHere Vij, kij, μij and fij denote, respectively, the normal velocity, the curvature, the mobility and the surface tension of the interface and the numbers Fij stand for the (constant) difference in bulk energies. At the point where the three phases coexist, local equilibrium requires thatIn case the material constants fij are small, and ε ≪ 1, previous analyses based on the parabolic nature of the equations (0.1) do not provide good qualitative information on the behaviour of solutions. In this case, it is more appropriate to consider the singular case with fij = 0. It turns out that this problem, (0.1) with fij = 0, admits infinitely many solutions. Here, we present results that strongly suggest that, in all cases, a unique solution—‘the vanishing surface tension (VST) solution’—is selected by letting ε→0. Indeed, a formal analysis of this limiting process motivates us to introduce the concept of weak viscosity solution for the problem with ε = 0. As we show, this weak solution is unique and is therefore expected to coincide with the VST solution. To support this statement, we present a perturbation analysis and a construction of self-similar solutions; a rigorous convergence result is established in the case of symmetric configurations. Finally, we use the weak formulation to write down a catalogue of solutions showing that, in several cases of physical relevance, the VST solution differs from results proposed previously.

Optical Detection of Laser-Induced Ionization: A Study of the Time Decay of Strontium Ions in the Air-Acetylene Flame

1986 ◽  
Vol 40 (8) ◽  
pp. 1085-1092 ◽  
Author(s):  
G. C. Turk ◽  
N. Omenetto
Keyword(s):  
State Transition ◽  
Electron Number Density ◽  
Dye Laser ◽  
Number Density ◽  
Fast Decay ◽  
Ion Chemistry ◽  
Electron Number ◽  
Slow Decay ◽  
Optical Probing ◽  
Density Of Electrons

Strontium atoms in the air-acetylene flame are directly photoionized in two steps provided by one dye laser tuned at the resonance ground-state transition (460.733 nm) and by the excimer pump beam at 308 nm, partially split from the amplifier section of the dye laser. The ions produced are then monitored by a third laser beam, colinear and counterpropagating in the flame, tuned to an ionic fluorescence transition and delayed in time with respect to the ionizing beams. In this way a fast decay, which is not affected by variations in the electron number density in the flame and therefore attributed to ion chemistry, and a slow decay, due to recombination, could clearly be observed. The fast decay is affected by variations in the flame stoichiometry and the slow decay by the number density of electrons in the flame, as shown by the addition of varying concentrations of an easily ionized element like caesium. The advantages of this optical probing of the laser-induced ionization in flames are discussed.

Asymptotic behaviour of positive solutions of semilinear elliptic problems with increasing powers

2021 ◽  
pp. 1-18
Author(s):  
Lucio Boccardo ◽  
Liliane Maia ◽  
Benedetta Pellacci
Keyword(s):  
Linear Operator ◽  
Asymptotic Behaviour ◽  
Positive Solutions ◽  
Elliptic Problems ◽  
Existence Results ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems ◽  
Image Position

We prove existence results of two solutions of the problem \[ \begin{cases} L(u)+u^{m-1}=\lambda u^{p-1} & \text{in}\ \Omega,\\ u>0 & \text{in}\ \Omega,\\ u=0 & \text{on}\ \partial \Omega, \end{cases} \] where $L(v)=-\textrm {div}(M(x)\nabla v)$ is a linear operator, $p\in (2,2^{*}]$ and $\lambda$ and $m$ sufficiently large. Then their asymptotical limit as $m\to +\infty$ is investigated showing different behaviours.

Perturbations of AF-Algebras

1979 ◽  
Vol 31 (5) ◽  
pp. 1012-1016 ◽  
Author(s):  
John Phillips ◽  
Iain Raeburn
Keyword(s):  
Hilbert Space ◽  
Unit Ball ◽  
Von Neumann Algebra ◽  
Unitary Operator ◽  
Affirmative Answer ◽  
Positive Answer ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Image Position ◽  
Af Algebras

Let A and B be C*-algebras acting on a Hilbert space H, and letwhere A1 is the unit ball in A and d(a, B1) denotes the distance of a from B1. We shall consider the following problem: if ‖A – B‖ is sufficiently small, does it follow that there is a unitary operator u such that uAu* = B?Such questions were first considered by Kadison and Kastler in [9], and have received considerable attention. In particular in the case where A is an approximately finite-dimensional (or hyperfinite) von Neumann algebra, the question has an affirmative answer (cf [3], [8], [12]). We shall show that in the case where A and B are approximately finite-dimensional C*-algebras (AF-algebras) the problem also has a positive answer.

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