Emden-Fowler Equation Research Articles

On variational methods to a generalized Emden–Fowler equation

In the present paper, the variational principle to the boundary value problems for a generalized Emden-Fowler equation is given and some existence results of solutions are obtained by using the critical point theory.

Oscillation criteria for a forced mixed type Emden–Fowler equation with impulses

Some oscillation criteria for a forced mixed type Emden–Fowler equation with impulses are given. When the impulses are dropped, our results extend those of Sun and Meng [Y.G. Sun, F.W. Meng, Interval criteria for oscillation of second-order differential equations with mixed nonlinearities, Appl. Math. Comput. 15 (2008) 375–381], Sun and Wong [Y.G. Sun, J.S.W. Wong, Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities, J. Math. Anal. Appl. 334 (2007) 549–560] for second-order forced ordinary differential equation with mixed nonlinearities, Nasr [A.H. Nasr, Sufficient conditions for the oscillation of forced superlinear second order differential equations with oscillatory potential, Proc. Am. Math. Soc. 126 (1998) 123–125], Yang [Q. Yang, Interval oscillation criteria for a forced second order nonlinear ordinary differential equations with oscillatory potential, Appl. Math. Comput. 135 (2003) 49–64] for forced superlinear Emden–Fowler equation, Kong [Q. Kong, Interval criteria for oscillation of second-order linear differential equations, J. Math. Anal. Appl. 229 (1999) 483–492] for unforced second order linear differential equations, and Wong [J.S.W. Wong, Oscillation criteria for a forced second order linear differential equation, J. Math. Anal. Appl. 231 (1999) 235–240] for forced second order linear differential equation.

On the Cauchy problem for an Emden-Fowler equation

We consider the equation y″ = P(x)xayσ, σ < 0, and prove the unique solvability of the Cauchy problem y(0) = 0, y′(0) = λ.

Sublinear eigenvalue problems on compact Riemannian manifolds with applications in Emden–Fowler equations

Let (M;g) be a compact Riemannian manifold without boundary, with dimM 3; and f : R! R a continuous function which is sublinear at innity. By various variational approaches, existence of multiple solutions of the eigenvalue problem g! + ( )! = ~ K(; )f(!); 2 M; !2 H 2 1 (M); is established for certain eigenvalues > 0, depending on further properties of f and on explicit forms of the function ~ K: Here, g stands for the Laplace{Beltrami operator on (M;g); and; ~ K are smooth positive functions. These multiplicity results are then applied to solve Emden{Fowler equations which involve sublinear terms at innity.

Existence of Solutions to Boundary Value Problems for the Discrete Generalized Emden-Fowler Equation

We consider the existence of solutions to boundary value problems for the discrete generalized Emden-Fowler equation. By means of the minimax methods in the critical point theory, some new results are obtained. Two examples are also given to illustrate the main results.

Nonoscillation criteria for second‐order nonlinear differential equations with decaying coefficients

AbstractThis paper is concerned with the problem of deciding conditions on the coefficient q (t) and the nonlinear term g (x) which ensure that all nontrivial solutions of the equation (|x ′|α–1x ′)′ + q (t)g (x) = 0, α &gt; 0, are nonoscillatory. The nonlinear term g (x) is not imposed no assumption except for the continuity and the sign condition xg (x) &gt; 0 if x ≠ 0. In our problem, it is important to examine the relation between the decay of q (t) and the growth of g (x). Our main result extends some nonoscillation theorem for the generalised Emden–Fowler equation. Proof is given by means of some Liapunov functions and phase‐plane analysis. A simple example is includes to show that the monotonicity of g (x) is not essential in our problem. Finally, elliptic equations with the m ‐Laplacian operator are discussed as an application to our results. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Positive solutions of the generalized Emden–Fowler equation in Hölder spaces

A singular sublinear BVP related to the Emden–Fowler equation is considered. Existence, nonexistence, and regularity of positive solutions in Hölder spaces is obtained.

The Korteweg–de Vries–Burgers equation and its approximate solution

In this paper, we study the Korteweg–de Vries–Burgers equation with boundary conditions. After making a series of transformations, we convert the Korteweg–de Vries–Burgers equation into an Emden–Fowler equation. An approximate solution in series form is obtained by means of the Adomian decomposition method. The solution is illustrated to agree well with phase plane analysis.

Multiple positive solutions to a singular boundary value problem for a superlinear Emden–Fowler equation

A multiplicity result for the singular ordinary differential equation y ″ + λ x −2 y σ = 0 , posed in the interval ( 0 , 1 ) , with the boundary conditions y ( 0 ) = 0 and y ( 1 ) = γ , where σ > 1 , λ > 0 and γ ⩾ 0 are real parameters, is presented. Using a logarithmic transformation and an integral equation method, we show that there exists Σ ⋆ ∈ ( 0 , σ / 2 ] such that a solution to the above problem is possible if and only if λ γ σ − 1 ⩽ Σ ⋆ . For 0 < λ γ σ − 1 < Σ ⋆ , there are multiple positive solutions, while if γ = ( λ −1 Σ ⋆ ) 1 / ( σ − 1 ) the problem has a unique positive solution which is monotonic increasing. The asymptotic behavior of y ( x ) as x → 0 + is also given, which allows us to establish the absence of positive solution to the singular Dirichlet elliptic problem − Δ u = d ( x ) −2 u σ in Ω, where Ω ⊂ R N , N ⩾ 2 , is a smooth bounded domain and d ( x ) = dist ( x , ∂ Ω ) .

Analytical-numerical investigation of a singular boundary value problem for a generalized Emden–Fowler equation

In this paper we are concerned about a singular boundary value problem for a quasilinear second-order ordinary differential equation, involving the one-dimensional p -laplacian. Asymptotic expansions of the one-parameter families of solutions, satisfying the prescribed boundary conditions, are obtained in the neighborhood of the singular points and this enables us to compute numerical solutions using stable shooting methods.

Oscillation Theorems Related to Averaging Technique for Second Order Emden-Fowler Type Neutral Differential Equations

Oscillation Theorems Related to Averaging Technique for Second Order Emden-Fowler Type Neutral Differential Equations

Determination of convergence intervals of the series solutions of Emden–Fowler equations using polytropes and isothermal spheres

We consider the Emden–Fowler type of equations. We construct a recurrence relation for the components of the approximate solution and investigate the convergence conditions. The previous results on the convergence radius of the series solution have been improved.

Multiple Positive Solutions in the Sense of Distributions of Singular BVPs on Time Scales and an Application to Emden-Fowler Equations

This paper is devoted to using perturbation and variational techniques to derive some sufficient conditions for the existence of multiple positive solutions in the sense of distributions to a singular second-order dynamic equation with homogeneous Dirichlet boundary conditions, which includes those problems related to the negative exponent Emden-Fowler equation.

Non-existence result for radially symmetric solutions to the Lane–Emden–Fowler equations

It is proved that the singular elliptic equation − Δ u = p ( x ) [ g ( u ) + f ( u ) + | ∇ u | q ] , u ( x ) > 0 in R N , u ( x ) → 0 as | x | → ∞ , where p ( x ) = p ( | x | ) ∈ C ( R N , ( 0 , ∞ ) ) , N > 2 , lim u ↘ 0 g ( u ) / u = lim u ↘ 0 f ( u ) / u = + ∞ , lim u ↗ ∞ g ( u ) / u = lim u ↗ ∞ f ( u ) / u = 0 , g ∈ C 1 ( ( 0 , ∞ ) , ( 0 , ∞ ) ) , f ∈ C l o c 0 , α ( [ 0 , ∞ ) , [ 0 , ∞ ) ) , 0 < α < 1 , has no positive radial solution that decays to zero near infinity provided ∫ 0 ∞ r p ( r ) d r = ∞ .

Blowing-up solutions to an anisotropic Emden–Fowler equation with a singular source

We consider the following anisotropic Emden–Fowler equation with a singular source − div ( a ( x ) ∇ v ) = ε 2 a ( x ) e v − 4 π α a ( p ) δ p in Ω , v = 0 on ∂ Ω , where p ∈ Ω ⊂ R 2 , constant α ∈ ( 0 , ∞ ) ∖ N , a ( x ) is a positive smooth function and δ p denotes the Dirac measure with pole at point p. If p is a local maximum point of a ( x ) , we construct a family of solutions v ε with arbitrary m bubbles concentrating at p, and the quantity ε 2 ∫ Ω a ( x ) e v ε → 8 π ( m + 1 + α ) a ( p ) .

Vertical asymptotes of solutions of the Emden-Fowler equation

Vertical asymptotes of solutions of the Emden-Fowler equation

Solutions of Emden–Fowler equations by homotopy-perturbation method

In this paper, approximate and/or exact analytical solutions of the generalized Emden–Fowler type equations in the second-order ordinary differential equations (ODEs) are obtained by homotopy-perturbation method (HPM). The homotopy-perturbation method (HPM) is a coupling of the perturbation method and the homotopy method. The main feature of the HPM is that it deforms a difficult problem into a set of problems which are easier to solve. In this work, HPM yields solutions in convergent series forms with easily computable terms, and in some cases, only one iteration leads to the high accuracy of the solutions. Comparisons with the exact solutions and the solutions obtained by the Adomian decomposition method (ADM) show the efficiency of HPM in solving equations with singularity.

The Supercritical Lane–Emden–Fowler Equation in Exterior Domains

We consider the exterior problem where 𝒟 is a bounded, smooth domain in ℝ N , for supercritical powers p > 1. We prove that if N ≥ 4 and , then this problem admits infinitely many solutions. If 𝒟 is symmetric with respect to N axes, this result holds whenever N ≥ 3 and .

Analysis of boundary bubbling solutions for an anisotropic Emden–Fowler equation

We consider the following anisotropic Emden–Fowler equation∇(a(x)∇u)+ɛ2a(x)eu=0in Ω,u=0on ∂Ω, where Ω⊂R2 is a smooth bounded domain and a is a positive smooth function. We study here the phenomenon of boundary bubbling solutions which do not exist for the isotropic case a≡ constant. We determine the localization and asymptotic behavior of the boundary bubbles, and construct some boundary bubbling solutions. In particular, we prove that if x¯∈∂Ω is a strict local minimum point of a, there exists a family of solutions such that ɛ2a(x)eudx tends to 8πa(x¯)δx¯ in D′(R2) as ɛ→0. This result will enable us to get a new family of solutions for the isotropic problem Δu+ɛ2eu=0 in rotational torus of dimension N⩾3.

Transformation between a generalized Emden–Fowler equation and the first Painlevé transcendent

AbstractWe use the generalized Sundman transformation to obtain a relation between a generalized Emden–Fowler equation and the first Painlevé transcendent. Copyright © 2007 John Wiley &amp; Sons, Ltd.