This paper is devoted to solving a multidimensional backward stochastic differential equation (BSDE for short) with a general random terminal time $τ$ taking values in $[0,+\infty]$. The generator $g$ of such BSDE satisfies a stochastic monotonicity condition in the state variable $y$ and a stochastic Lipschitz condition in the state variable $z$ with possibly unbounded stochastic coefficients $μ_\cdot\in\R$ and $ν_\cdot\in\R_+$ satisfying $\int_0^τ(|μ_t|+ν^2_t) {\rm d}t<+\infty$, along with a very general growth in $y$ that is more easily verified and weaker than existing ones. Let $p>1$ be a given constant and $ρ_\cdot\geq μ_\cdot+\fracθ{2[1\wedge(p-1)]}ν_\cdot^2$ be a given real-valued process for some constant $θ>1$ such that $\int_0^τ|ρ_t|{\rm d}t<+\infty$. In a general weighted $L^p$ space with a weighted factor $e^{\int_0^t ρ_r{\rm d}r}$, we establish an existence and uniqueness result for the adapted solution of previous BSDE when the terminal value satisfies an associated weighted integrability condition, broadening the scope of the process $ρ_\cdot$ in the weighted factor and thereby unifying and strengthening some corresponding existing results obtained in \citet{DarlingandPardoux1997}, \citet{Briand2003}, \citet{LiFan2024SD} and \citet{Li2025}. Some innovative ideas are presented in order to address the general weighted space and the very general growth condition. As applications, we prove the existence of viscosity solutions for parabolic and elliptic PDEs linked with previous BSDEs under some general assumptions on their nonlinear terms, and establish a dual representation of an unbounded dynamic concave utility defined on a general weighted $L^p$ space via the weighted $L^p$ solutions of previous BSDEs.