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June 1985
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217 Reads
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35 Citations
Most known conditions implying that a nonlinear differential equation y ( n )+ f(t, y) = 0 has solutions such that lim t →∞ t − k y ( i ) = c ≠ 0 are local in that the solutions are guaranteed to exist only for sufficiently large t . This paper presents conditions ensuring that the solutions exist on a given interval and have the prescribed asymptotic behaviour. Some of the integral smallness conditions on f permit conditional convergence.
January 1985
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32 Reads
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45 Citations
Annali di Matematica Pura ed Applicata
Conditions are given for the nonlinear differential equation (1)L n y+f(t, y, ..., ...,y (n–1)=0to have solutions which exist on a given interval [t0, )and behave in some sense like specified solutions of the linear equation (2)L n z=0as t.The global nature of these results is unusual as compared to most theorems of this kind, which guarantee the existence of solutions of (1)only for sufficiently large t. The main theorem requires no assumptions regarding oscillation or nonoscillation of solutions of (2).A second theorem is specifically applicable to the situation where (2)is disconjugate on [t 0, ),and a corollary of the latter applies to the case where Lz=z n.
January 1985
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1 Read
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2 Citations
Mathematische Nachrichten
January 1985
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3 Reads
Proceedings of the American Mathematical Society
The equation ( ∗ ) Δ u = ϕ ( x ) u γ ( * )\Delta u = \phi (x){u^\gamma } is considered in R 2 {{\mathbf {R}}^2} , where γ ≠ 1 \gamma \ne 1 and ϕ ( x ) ⩾ 0 \phi (x) \geqslant 0 is locally Hölder continuous. Sufficient conditions are obtained for ( ∗ ) ( * ) to possess infinitely many positive solutions which are defined throughout R 2 {R^2} and have logarithmic growth as | x | → ∞ |x| \to \infty . An extension of the main result to the higher-dimensional case is also attempted.
January 1985
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16 Reads
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53 Citations
Indiana University Mathematics Journal
January 1985
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12 Reads
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19 Citations
Proceedings of the American Mathematical Society
The equation (*) Δ u = φ(x)uγ is considered in R2, where γ ≠ 1 and φ(x) ≥ 0 is locally Hölder continuous. Sufficient conditions are obtained for (*) to possess infinitely many positive solutions which are defined throughout R2 and have logarithmic growth as |x| → ∞. An extension of the main result to the higher-dimensional case is also attempted.
January 1985
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20 Reads
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84 Citations
Japanese journal of mathematics
September 1984
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1 Read
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39 Citations
Mathematische Zeitschrift
On considere des systemes faiblement couples d'equations elliptiques semilineaires d'ordre 2 de la forme: Δu+Φ(x,u,v)=0; Δv+Ψ(x,u,v)=0 dans R n , n≥3. On developpe une theorie de l'existence des solutions entieres pour ce systeme
June 1984
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3 Reads
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10 Citations
Canadian mathematical bulletin = Bulletin canadien de mathématiques
Necessary and sufficient conditions are found for the existence of two positive solutions of the semilinear elliptic equation Δ u + q (|x|) u = f ( x , u ) in an exterior domain Ω⊂ℝ ⁿ , n ≥ 1, where q , f are real-valued and locally Hölder continuous, and f ( x , u ) is nonincreasing in u for each fixed x ∈Ω. An example is the singular stationary Klein-Gordon equation Δ u — k ² u = p ( x ) u -λ where k and λ are positive constants. In this case NASC are given for the existence of two positive solutions u i ( x ) in some exterior subdomain of Ω such that both |x| m exp[(-l) i-1 k | x |] u i ( x ) are bounded and bounded away from zero in this subdomain, m = ( n —1)/2, i = 1, 2.
June 1984
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4 Reads
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25 Citations
Mathematische Annalen
On considere l'equation elliptique a 2 dimensions Δu=f(x,u,⊇u), x∈R 2 , ou x=(x 1 ,x 2 ) et f(x,u,p) est une fonction continue non negative sur R 2 ×R + XR 2 . On considere l'existence de solutions positives definies dans tout R 2
... It is also important to observe that equation of the form (1) arises naturally in the study of radially symmetric solutions (ground states) of semi-linear equations, and many works have been conducted in this area; see [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. ...
January 1985
Japanese journal of mathematics
... = D° j^) > 0 for r ^ r^-. Thus we again obtain a contradiction to the oscillation of all nontrivial solutions of (11), because the function w(r) = z(r) + a x p 0 (r) is an eventually positive solution of (11). ...
January 1983
Archivum Mathematicum
... In this regard, see [84] for an extensive complete study of a specific nonlinear equation and [85] for a bibliographical study of unforced equations of the form y ′′ (x) + F (x, y(x), y ′ (x)) = 0. For results which compare the non-oscillatory behavior of forced equations of the form (1) with those of the associated unforced equation, (6) below, and possible equations with delays, we refer the reader to [1], [33] and [73]. One should not forget that even though the literature is filled with sufficient criteria for oscillation/non-oscillation of unforced equations like ...
December 1982
Journal of Mathematical Analysis and Applications
... Due first to the fact that q may change signs and secondly to the presence of the function ф our results difibr from those previously obtained for equations (El) and (E2) respectively. Moreover, our results will cover all solutions not just the bounded ones (for example, see some of the results in [3] and [11]). In addition, we do not require J°° q(s) ds = 00 as many authors do (see some of the results in [3], [11], or [17]), and in that respect even when i^(x) = 1, our results differ from some of those previously known for equation (E2). ...
December 1974
Hiroshima Mathematical Journal
... The asymptotic integration problem for second-order ordinary differential equations is a classical research topic in mathematics. It has been widely investigated by many authors for the last several decades, see for instance [1][2][3][4][5][6][7][8][9][10] and the references cited therein. The problem is to find sufficient conditions to guarantee the existence of a solution with a prescribed behavior at infinity. ...
January 1986
Hiroshima Mathematical Journal
... where $r_{0}>0$ is a sufficiently large number. When $f(u)=|u|'-1u$, $y$ $>0,$ oscillation criteria for (1), which can be regarded as generalizations of earilier results in $[1, 5]$ , have been obtained in [4]. The case of $m$ $=2$ has been treated in [3]. ...
January 1978
Utilitas Mathematica
... , [15]) . The existence of positive solutions of the equation on exterior domains (including R n ) has been widely considered (for example, see [3], [4], [7], [10], [11], [14], and references therein). The main approach used to prove existence results is to construct super and sub solutions. ...
January 1983
Funkcialaj Ekvacioj
... Oscillation criteria obtained by Kusano andOnose 1973 andby Belohorec 1969 are extended to second-order sublinear impulsive differential equations of Emden-Fowler type: x t p t |x τ t | α−1 x τ t 0, t / θ k ; Δx t | t θk q k |x τ θ k | α−1 x τ θ k 0; Δx t | t θk 0, 0 < α < 1 by considering the cases τ t ≤ t and τ t t, respectively. Examples are inserted to show how impulsive perturbations greatly affect the oscillation behavior of the solutions. ...
September 1973
Proceedings of the American Mathematical Society
... The study of oscillation of higher order nonlinear functional differential equations with deviating arguments was attempted for the first time by Onose [28,29]. A typical generalization of Onose's oscillation theorem can be found in [11]. Recently, wide attention of the researchers has been attracted to the investigation of oscillation (or nonoscillation) of differential equations whose principal differential operators involve nonlinear Sturm-Liouville type differential operators [12, 20, 22, 30 -34]. ...
September 1980
Journal of Mathematical Analysis and Applications
... were obtained in [6,11,13,14] and [25]. Here, λ, τ > 0 and ω ± (t + τ ) = ∓ω ± (t). ...
January 1995
Mathematica Bohemica
