| Abstract: |
| We present the existence, uniqueness, and sharp bilateral pointwise estimates for positive bounded solutions to the Lane--Emden type problem
\[
\begin{cases}
\mathcal{L} u = \sum\limits_{i=1}^{m}\sigma_{i} u^{q_{i}}+\sigma_0, \quad u\geq0 & \text{in} \; \Omega, \
\liminf \limits_{x \rightarrow y} u(x) = f(y), & y \in \partial^\infty\Omega, \end{cases}
\] where $0 < q_{i} < 1$.
Here $\mathcal{L} u = - \operatorname{div}(\mathbb A \nabla u)$ is a uniformly elliptic operator with bounded coefficients,
$\sigma_{i}$ is a nonnegative locally finite Borel measure on an $\mathbb A$-regular domain $\Omega \subset \mathbb R^n$ which
possesses a positive Green function associated with $\mathcal{L}$, and $f$ is a nonnegative continuous function on the
boundary $\partial^\infty\Omega$. An analogous result for positive continuous solutions to the problem is also illustrated.
Our method can be adapted to address related sublinear problems with zero boundary conditions involving the fractional
Laplace operator $(-\Delta)^{\alpha}$ for $0< \alpha < n/2$, in place of $\mathcal{L}$, in $\mathbb{R}^n$ as well. This is a
joint work with Kentaro Hirata (Hiroshima) and Toe Toe Shwe (SIIT). |
|