Radial entire solutions to even order semilinear elliptic equations in the plane

1987 ◽  
Vol 105 (1) ◽  
pp. 275-287 ◽  
Author(s):  
Takaŝi Kusano ◽  
Manabu Naito ◽  
Charles A. Swanson
Keyword(s):  
Elliptic Equations ◽  
Sufficient Conditions ◽  
Semilinear Elliptic Equations ◽  
Two Dimensional ◽  
Entire Solutions ◽  
Semilinear Elliptic ◽  
Negative Constant ◽  
Even Order ◽  
Image Position ◽  
Necessary And Sufficient

SynopsisSemilinear elliptic equations of the formare considered, where Δm is the m-th iterate of the two-dimensional Laplacian Δ, p(t) is continuous in [0, ∞), and f(u is continuous and positive either in (0, ∞) or in ℝ.Our main objective is to present conditions on p and f which imply the existence of radial entire solutions to (*), that is, those functions of class C2m(ℝ2) which depend only on |x| and satisfy (*) pointwise in ℝ2.First, necessary and sufficient conditions are established for equation (*), with p(t) > 0 in [0, ∞), to possess infinitely many positive radial entire solutions which are asymptotic to positive constant multiples of |x|2m−2 log |x as |x| → ∞. Secondly, it is shown that, in the case p(t < 0, in [ 0, ∞) and f(u) > 0 is nondecreasing in ℝ, equation (*) always has eventually negative radial entire solutions, all of which decrease at least as fast as negative constant multiples of |x|2m−2 log |x| as |x| → ∞. Our results seem to be new even when specialised to the prototypeswhere γ is a constant.

Radial Entire Solutions of Even Order Semilinear Elliptic Equations

1988 ◽  
Vol 40 (6) ◽  
pp. 1281-1300 ◽  
Author(s):  
Takaŝi Kusano ◽  
Manabu Naito ◽  
Charles A. Swanson
Keyword(s):  
Partial Differential Equations ◽  
Continuous Function ◽  
Positive Integer ◽  
Positive Constant ◽  
Elliptic Equations ◽  
Elliptic Partial Differential Equations ◽  
Entire Solutions ◽  
Semilinear Elliptic ◽  
Even Order ◽  
Image Position

Semilinear elliptic partial differential equations of the type1will be considered throughout real Euclidean N-space, where m ≧ 2 is a positive integer, Δ denotes the N-dimensional Laplacian, and f is a real-valued continuous function in [0, ∞) × (0, ∞). Detailed hypotheses on the structure of f are listed in Section 3.Our objective is to prove the existence of radially symmetric positive entire solutions u(x) of (1) which are asymptotic to positive constant multiples of |x|2m−2i as |x| → ∞ for every i = 1,…, m, N ≧ 2i + 1.

Multiple positive solutions of nonhomogeneous semilinear elliptic equations in ℝN

1996 ◽  
Vol 126 (2) ◽  
pp. 443-463 ◽  
Author(s):  
Cao Dao-Min ◽  
Zhou Huan-Song
Keyword(s):  
Positive Solutions ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Multiple Positive Solutions ◽  
Semilinear Elliptic ◽  
Sobolev Constant ◽  
Image Position

We consider the following problemwhere for all ≦f(x,u)≦c1up-1 + c2u for all x ∈ℝN,u≧0 with c1>0,c2∈(0, 1), 2<p<(2N/(N – 2)) if N ≧ 3, 2 ≧ + ∝ if N = 2. We prove that (*) has at least two positive solutions ifand h≩0 in ℝN, where S is the best Sobolev constant and

Characterization of isochronous foci for planar analytic differential systems

2005 ◽  
Vol 135 (5) ◽  
pp. 985-998 ◽  
Author(s):  
Jaume Giné ◽  
Maite Grau
Keyword(s):  
Critical Point ◽  
Analytic Functions ◽  
Autonomous Systems ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Differential Systems ◽  
Systems Of Differential Equations ◽  
Image Position ◽  
Necessary And Sufficient

We consider the two-dimensional autonomous systems of differential equations of the form where P(x,y) and Q(x,y) are analytic functions of order greater than or equal to 2. These systems have a focus at the origin if λ ≠ 0, and have either a centre or a weak focus if λ = 0. In this work we study the necessary and sufficient conditions for the existence of an isochronous critical point at the origin. Our result is, to the best of our knowledge, original when applied to weak foci and gives known results when applied to strong foci or to centres.

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