We are concerned with the existence of positive solutions to the following boundary value problem in \((0,\infty ),\)$$\begin{aligned} \frac{1}{A}\left( A\phi \left( \left| u^{\prime }\right| \right) u^{\prime }\right) ^{\prime }=-a(t)u^{\alpha },t>0,\left( A\phi \left( \left| u^{\prime }\right| \right) u^{\prime }\right) \left( 0\right) =0\text { and}\lim \nolimits _{t\rightarrow +\infty }u(t)=0, \end{aligned}$$where \(\alpha \ge 0,\)\(\phi \) is a nonnegative continuously differentiable function on \(\left[ 0,\infty \right) \), A is a continuous function on \( \left[ 0,\infty \right) \), differentiable, positive on \(\left( 0,\infty \right) \) and a is a nonnegative function satisfying some appropriate assumptions related to Karamata regular variation theory. We give also, estimates on such solutions.

中文翻译：

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关于涉及$$ \ Phi $$Φ-Laplacian的问题的基态径向解的渐近行为

我们关心以下\（（0，\ infty），\）$$ \ begin {aligned} \ frac {1} {A} \ left（A \ phi \ left （\ left | u ^ {\ prime} \ right | \ right）u ^ {\ prime} \ right）^ {\ prime} =-a（t）u ^ {\ alpha}，t> 0，\ left（ A \ phi \ left（\ left | u ^ {\ prime} \ right | \ right）u ^ {\ prime} \ right）\ left（0 \ right）= 0 \ text {and} \ lim \ nolimits _ { t \ rightarrow + \ infty} u（t）= 0，\ end {aligned} $$其中\（\ alpha \ ge 0，\）\（\ phi \）是\（\ left [ 0，\ infty \右）\） ，阿是一个连续函数\（\左[0，\ infty \右）\） ，微，正上\（\左（0，\ infty \右）\）和a是一个非负函数，满足与Karamata正变分理论相关的一些适当假设。我们还给出了此类解决方案的估算值。