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The existence of decaying positive solutions in ${ℝ}_{+}$ of the equations $\left(E_{\lambda }\right)$ and $\left(E_{\lambda }^1\right)$ displayed below is considered. From the existence of such solutions for the subhomogeneous cases (i.e. $t^{1-p}F(r,tU,t|U^{\text{'}}\left|\right)\searrow 0$ as $t\nearrow \infty$), a super-sub-solutions method (see § 2.2) enables us to obtain existence theorems for more general cases.

In this paper, the axisymmetric flow in an ideal fluid outside the infinite cylinder ($r\le d$) where $(r,\theta ,z)$ denotes the cylindrical co-ordinates in ${ℝ}^3$ is considered. The motion is with swirl (i.e. the $\theta$-component of the velocity of the flow is non constant). The (non-dimensional) equation governing the phenomenon is (Pd) displayed below. It is known from e.g. that for the problem without swirl ($f_q=0$ in (f)) in the whole space, as the flux constant $k$ tends to $\infty$, 1) $\mathrm{dist}(0z,\partial A)=O\left(k^{1/2}\right)$; $\mathrm{diam}A=O(exp(-c_0{k}^{3/2}))$; 2) ${\left(k^{1/2}\Psi \right)}_{k\in \mathbb{N}}$ converges to a vortex cylinder $U_m$ (see...