In this study, we consider the following quasilinear Choquard equation with singularity − Δ u + V ( x ) u − u Δ u 2 + λ ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u = K ( x ) u − γ , x ∈ R N , u > 0 , x ∈ R N , \left\{\begin{array}{ll}-\Delta u+V\left(x)u-u\Delta {u}^{2}+\lambda \left({I}_{\alpha }\ast | u{| }^{p})| u{| }^{p-2}u=K\left(x){u}^{-\gamma },\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\\ u\gt 0,\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\end{array}\right. where I α {I}_{\alpha } is a Riesz potential, 0 < α < N 0\lt \alpha \lt N , and N + α N < p < N + α N − 2 \displaystyle \frac{N+\alpha }{N}\lt p\lt \displaystyle \frac{N+\alpha }{N-2} , with λ > 0 \lambda \gt 0 . Under suitable assumption on V V and K K , we research the existence of positive solutions of the equations. Furthermore, we obtain the asymptotic behavior of solutions as λ → 0 \lambda \to 0 .

中文翻译：

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具有奇异性的拟线性Choquard方程解的存在性和渐近性

在这项研究中，我们考虑以下具有奇异性的拟线性Choquard方程-Δu + V（x）u-uΔu 2 +λ（Iα∗ ∣ u ∣ p）∣ u ∣ p − 2 u = K（x） u −γ，x∈RN，u> 0，x∈RN，\ left \ {\ begin {array} {ll}-\ Delta u + V \ left（x）uu \ Delta {u} ^ {2} + \ lambda \ left（{I} _ {\ alpha} \ ast | u {|} ^ {p}）| 你{| } ^ {p-2} u = K \ left（x）{u} ^ {-\ gamma}，\ hspace {1.0em}＆x \ in {{\ mathbb {R}}} ^ {N}，\ \ u \ gt 0，\ hspace {1.0em}＆x \ in {{\ mathbb {R}}} ^ {N}，\ end {array} \ right。其中Iα{I} _ {\ alpha}是Riesz势，0 <α<N 0 \ lt \ alpha \ lt N，并且N +αN <p <N +αN-2 \ displaystyle \ frac {N + \ alpha} {N} \ lt p \ lt \ displaystyle \ frac {N + \ alpha} {N-2}，其中λ> 0 \ lambda \ gt 0。在对VV和KK的适当假设下，我们研究了方程正解的存在性。此外，