• Kyungpook Mathematical Journal
• /
• v.46 no.3
• /
• pp.329-336
• /
• 2006

In the category of the group theoretic methods using invertible discrete group transformation, we give a useful relation between Emden-Fowler equations and nonlinear heat equation. In this paper, by means of appropriate transformations of discrete group analysis, the nonlinear hate equation transformed into the class of the Emden-Fowler equations. This approach shows that, under the group action, the solution of reference equation can be transformed into the solution of the transformed equation.

• Kyungpook Mathematical Journal
• /
• v.64 no.1
• /
• pp.113-131
• /
• 2024

In this paper we give conditions for the existence of, and describe the asymtotic behavior of, radial positive solutions of the nonlinear elliptic equation of Emden-Fowler type with convection term ∆_p u + 𝛼|u|^q-1u + 𝛽x.∇(|u|^q-1u) = 0 for x ∈ ℝ^N, where p > 2, q > 1, N ≥ 1, 𝛼 > 0, 𝛽 > 0 and ∆_p is the p-Laplacian operator. In particular, we determine ${\lim}_{r{\rightarrow}}{\infty}\,r^{\frac{p}{q+1-p}}\,u(r)$ when $\frac{{\alpha}}{{\beta}}$ > N > p and $q\,{\geq}\,{\frac{N(p-1)+p}{N-p}}$.

• Journal of the Korean Mathematical Society
• /
• v.37 no.5
• /
• pp.763-777
• /
• 2000

We investigate the existence of group invariant solutions of the Emden-Fowler equation, - u=$\mid$x$\mid$$\sigma$$\mid$u$\mid$p-1u in B, u=0 on B and u(gx)=u(x) in B for g G. Here B is the unit ball in n 2, 1$\sigma$ 0 and G is a closed subgrop of the orthogonal group. A soultion of the problem is called a G in variant solution. We prove that there exists a G invariant non-radial solution if and only if G is not transitive on the unit sphere.

• Journal of applied mathematics & informatics
• /
• v.28 no.5_6
• /
• pp.1285-1304
• /
• 2010

The purpose of this paper is to establish some sufficient conditions for oscillation of solutions of the second-order functional dynamic equation of Emden-Fowler type $\[a(t)x^{\Delta}(t)\]^{\Delta}+p(t)|x^{\gamma}(\tau(t))|\|x^{\Delta}(t)\|^{1-\gamma}$ $sgnx(\tau(t))=0$, $t\;{\geq}\;t_0$, on a time scale $\mathbb{T}$, where ${\gamma}\;{\in}\;(0,\;1]$, a, p and $\tau$ are positive rd-continuous functions defined on $\mathbb{T}$, and $lim_{t{\rightarrow}{\infty}}\;{\tau}(t)\;=\;\infty$. Our results include some previously obtained results for differential equations when $\mathbb{T}=\mathbb{R}$. When $\mathbb{T}=\mathbb{N}$ and $\mathbb{T}=q^{\mathbb{N}_0}=\{q^t\;:\;t\;{\in}\;\mathbb{N}_0\}$ where q > 1, the results are essentially new for difference and q-difference equations and can be applied on different types of time scales. Some examples are worked out to demonstrate the main results.