Dirichlet problem for Lane-Emden type equations with several sublinear terms

TL;DR

This paper establishes existence, uniqueness, and pointwise estimates for positive solutions to a class of Lane-Emden equations with sublinear terms involving elliptic operators and measures.

Contribution

It provides new results on positive solutions to sublinear elliptic equations with measure data and extends methods to fractional Laplacian problems.

Findings

Proved existence and uniqueness of solutions.

Derived sharp bilateral pointwise estimates.

Extended results to fractional Laplace operators.

Abstract

We prove the existence, uniqueness, and sharp bilateral pointwise estimates for positive bounded solutions to the Lane--Emden type problem \[ \begin{cases} L u = \sum\limits_{i=1}^{m}\sigma_{i} u^{q_{i}}+\sigma_0, \quad u\geq0 & \text{in } \Omega, \liminf \limits_{x \rightarrow y} u(x) = f(y), & y \in \partial^\infty\Omega, \end{cases} \] where $0 < q_{i} < 1$. Here $Lu = - \text{div}(A \nabla u)$ is a uniformly elliptic operator with bounded coefficients, $\sigma_{i}$ is a nonnegative locally finite Borel measure on an $A$-regular domain $\Omega \subset \mathbb{R}^n$ which possesses a positive Green function associated with $L$, 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 $L$, in $\mathbb{R}^n$ as well.