| Abstract: |
| We will focus on the asymptotic behavior of the solutions to the boundary value problem
$$ -\Delta_p u -\Delta_q u = \lambda g(x)\ \mbox{in}\ \Omega$$
and
$$ u =0\ \mbox{on}\ \partial \Omega$$
as $\lambda$ approaches b $\infty$ where $\Omega$ in a smooth bounded domain in $\mathbb{R}^N$ and $g: \Omega \rightarrow \mathbb{R}$ is indefinite in sign and possibly singular near the boundary of $\Omega.$ These estimates find application in establishing the existence of a positive solution to a related problem
$$ -\Delta_p u -\Delta_q u = \lambda f(u)\ \mbox{in}\ \Omega$$
with zero boundary conditions. Here we consider the non-linearity $f:(0,\infty) \rightarrow \mathbb{R}$ to be $p$-sublinear at infinity. Moreover, when $f$ takes a specific form, we obtain a positive solution that also serves as a local minimizer for the associated energy functional. |
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