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} -Δ_{p} u + (-Δ)^{s}_{q} u = f(x) u^{-α} + g(x) u^β, \quad u > 0 \quad $\text{in }$ Ω; \quad u = 0, \quad $\text{in }$ $\mathbb{R}$^{N} \setminus Ω, \end{equation} where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is an open bounded domain with a $C^2$ boundary $\mathrm{\partial }\mathrm{\Omega }$, and $N>p$. We assume that $0<s<1$ and $1<p,q<\mathrm{\infty }$, 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 $L^r(\mathrm{\Omega })$ for some $r\in [1,\mathrm{\infty }]$, or satisfies a growth condition involving negative powers of the distance function $d(\cdot )$ near the boundary $\mathrm{\partial }\mathrm{\Omega }$. 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 $\delta \ge p$. Our approach leverages the Picone identities on one hand and the interaction between the local and non-local terms on the other hand.