About the radial solutions of the nonlinear $(p_1,p_2)$-Laplacian problem

Let $p_1, p_2>1$, we consider the following sum of two different $p$-Laplacians problem
$$\begin{cases}-L_{p_1}u-L_{p_2}u=a(t)u^{\sigma}\quad \text{on}\ \ (0,1), \\ \displaystyle\lim_{t \longrightarrow 0} A(t)\bigl(| u' |^{p_1-2}u' +|u'|^{p_2-2}u'\bigr) (t)=0, \\ u(1)=0, \end{cases}$$
where $0<\sigma<\min(p_1, p_2)-1$ and the operator $L_{p}u$ is defined by
$$ L_{p}u:=\dfrac{1}{A}(A| u' |^{p-2} u')' $$
for $p>1$. We provide sufficient conditions on the functions $A$ and $a$ that yield the existence, and we give the asymptotic behavior of radial positive solutions. An example is given to illustrate the applicability of our main results.