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Abstract Consider the following stochastic differential equation (SDE): X t = x + ∫ 0 t b ( s , X s ) d s + ∫ 0 t σ ( s , X s ) d B s , 0 ≤ t ≤ T , x ∈ R , $$ X_{t}=x+ \int _{0}^{t}b(s,X_{s})\,ds+ \int _{0}^{t}\sigma (s,X_{s}) \,dB_{s}, \quad 0\leq t\leq T, x\in \mathbb{R}, $$ where { B s } 0 ≤ s ≤ T $\{B_{s}\}_{0\leq s\leq T}$ is a 1-dimensional standard Brownian motion on [ 0 , T ] $[0,T]$ . Suppose that q ∈ ( 1 , ∞ ] $q\in (1,\infty ]$ , p ∈ ( 1 , ∞ ) $p\in (1,\infty )$ , b = b 1 + b 2 $b=b_{1}+b_{2}$ , b 1 ∈ L q ( 0 , T ; L p ( R ) ) $b_{1}\in L^{q}(0,T;L^{p}(\mathbb{R}))$ such that 1 / p + 2 / q < 1 $1/p+2/q<1$ and b 2 $b_{2}$ is bounded measurable, with σ ∈ L ∞ ( 0 , T ; C u ( R ) ) $\sigma \in L^{\infty }(0,T;{\mathcal{C}}_{u}(\mathbb{R}))$ there being a real number δ > 0 $\delta >0$ such that σ 2 ≥ δ $\sigma ^{2}\geq \delta $ . Then there exists a weak solution to the above equation. Moreover, (i) if σ ∈ C ( [ 0 , T ] ; C u ( R ) ) $\sigma \in \mathcal{C}([0,T];\mathcal{C}_{u}(\mathbb{R}))$ , all weak solutions have the same probability law on 1-dimensional classical Wiener space on [ 0 , T ] $[0,T]$ and there is a density associated with the above SDE; (ii) if b 2 = 0 $b_{2}=0$ , p ∈ [ 2 , ∞ ) $p\in [2,\infty )$ and σ ∈ L 2 ( 0 , T ; C b 1 / 2 ( R ) ) $\sigma \in L^{2}(0,T;{\mathcal{C}}_{b}^{1/2}({\mathbb{R}}))$ , the pathwise uniqueness holds.