Uniqueness Results for Mixed Local and Nonlocal Equations with Singular Nonlinearities and Source Terms

This paper considers a local and non-local problem characterized by singular nonlinearity and a source term. Specifically, we focus on the following problem: \begin{equation}\label{A}\tag{P} -\Delta_{p} u + (-\Delta)^{s}_{q} u = f(x) u^{-\alpha} + g(x) u^{\beta}, \quad u > 0 \quad \text{in } \Omega; \quad u = 0, \quad \text{in } \mathbb{R}^{N} \setminus \Omega, \end{equation} where $\Omega \subset \mathbb{R}^N$ is an open bounded domain with a $C^{2}$ boundary $\partial \Omega$, and $N > p$. We assume that $0 < s < 1$ and $1 < p, q < \infty$, with the conditions $q = p$ or $q < p$, corresponding to the homogeneous and non-homogeneous cases, respectively. The parameters satisfy $0 < \beta < q - 1$ and $\alpha > 0$. The function $f$ is non-zero and belongs to a suitable Lebesgue space $L^{r}(\Omega)$ for some $r \in [1, \infty]$, or satisfies a growth condition involving negative powers of the distance function $d(\cdot)$ near the boundary $\partial \Omega$. Additionally, $g$ is a nonnegative function within appropriate Lebesgue spaces. The primary objectives of this paper are twofold. First, we establish the uniqueness of infinite energy solutions to problem \eqref{A} by introducing a novel comparison principle under certain conditions. Second, we derive several existence results for weak solutions in various senses, accompanied by regularity results for problem \eqref{A}. Furthermore, we present a non-existence result when the function $f(x) \sim d^{-\delta}(x)$ and $x$ is near the boundary, under the condition $\delta \geq p$. Our approach leverages the Picone identities on one hand and the interaction between the local and non-local terms on the other hand.