Asymptotic behavior of ground state radial solutions for problems involving the $\Phi$-Laplacian

Abstract

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.