Abstract
Via an integral transformation, we establish two embedding results between the Emden-Fowler type equation \({\frac{{\rm d}^{2}x}{{\rm d}t^{2}}=a(t)x^{\lambda}}\) , t ≥ t 0 > 0, with solutions x such that \({x(t)\sim c\cdot t^{2}}\) as \({t\rightarrow +\infty}\) , \({c\not= 0}\) , and the equation \({\frac{{\rm d}^{2}y}{{\rm d}u^{2}}=b(u)e^{y}}\) , u > 0, with solutions y such that \({\lim sup_{u\searrow k}\frac{y(u)}{\ln (u-k)}=c_{1} < 0}\) for given k > 0. The conclusions of our investigation are used to derive conditions for the existence of radial solutions to the elliptic equation \({\Delta U=K(\left| x\right|)e^U}\) , \({\left| x\right| > x_0 > 0}\) , that blow up as \({\left| x\right|\searrow x_0}\) in the two dimensional case.
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Mustafa, O.G. On t 2-like solutions of certain second order differential equations. Annali di Matematica 187, 187–196 (2008). https://doi.org/10.1007/s10231-007-0040-7
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DOI: https://doi.org/10.1007/s10231-007-0040-7