Year: 2026
Type: preprint
Abstract: We prove the existence, uniqueness, and sharp bilateral pointwise estimates for positive bounded solutions to the Lane--Emden type problem \[ \begin{cases} L u = \sum\limits_{i=1}^{m}σ_{i} u^{q_{i}}+σ_0, \quad u\geq0 & \text{in } Ω, \liminf \limits_{x \rightarrow y} u(x) = f(y), & y \in \partial^\inftyΩ, \end{cases} \] where $0 < q_{i} < 1$. Here $Lu = - \text{div}(A \nabla u)$ is a uniformly elliptic operator with bounded coefficients, $σ_{i}$ is a nonnegative locally finite Borel measure on an $A$-regular domain $Ω\subset \mathbb{R}^n$ which possesses a positive Green function associated with $L$, and $f$ is a nonnegative continuous function on the boundary $\partial^\inftyΩ$. 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 $(-Δ)^α$ for $0< α< n/2$, in place of $L$, in $\mathbb{R}^n$ as well.
Source: arXiv (Cornell University)
Institution: Thammasat University
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7160852270
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