Uniqueness results for mixed local and nonlocal equations with singular nonlinearities and source terms

This paper considers a local and nonlocal problem characterized by singular nonlinearity and a source term. Specifically, we focus on the following problem: P −Δpu+(−Δ)qsu=f(x)u−α+g(x)uβ,u>0inΩ;u=0,inRN∖Ω,$$\begin{equation} -\Delta _{p} u + (-\Delta)^{s}_{q} u = f(x) u^{-\alpha } + g(x) u^{\beta }, \quad u > 0 \quad \text{in } \Omega; \quad u = 0, \quad \text{in } \mathbb {R}^{N} \setminus \Omega, \end{equation}$$where Ω⊂RN$ \Omega \subset \mathbb {R}^N$ is an open bounded domain with a C2$ C^{2}$ boundary ∂Ω,$ \partial \Omega,$ and N>p.$ N > p.$ We assume that 0 0$ \alpha > 0$. The function f$ f$ is nonzero and belongs to a suitable Lebesgue space Lr(Ω)$ L^{r}(\Omega)$ for some r∈[1,∞]$ r \in [1, \infty]$, or satisfies a growth condition involving negative powers of the distance function d(·)$ d(\cdot)$ near the boundary ∂Ω$ \partial \Omega$. Additionally, g$ g$ is a positive function defined within appropriate Lebesgue spaces. The primary objectives of this paper are twofold. First, we establish the uniqueness of infinite energy solutions to problem (P) 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 (P). Furthermore, we present a nonexistence result when the function f(x)∼d−δ(x)$ f(x) \sim d^{-\delta }(x)$ and x$ x$ is near the boundary, under the condition δ≥p$ \delta \ge p$. Our approach leverages the Picone identities on one hand and the interaction between the local and nonlocal terms on the other hand.