scholarly journals Isolated singularities of positive solutions for Choquard equations in sublinear case

2018 ◽  
Vol 20 (04) ◽  
pp. 1750040 ◽  
Author(s):  
Huyuan Chen ◽  
Feng Zhou
Keyword(s):  
Positive Solutions ◽  
Riesz Potential ◽  
Singular Solutions ◽  
Nonlocal Term ◽  
Choquard Equation ◽  
Qualitative Properties ◽  
Isolated Singularities ◽  
Nonexistence And Existence

Our purpose of this paper is to study the isolated singularities of positive solutions to Choquard equation in the sublinear case [Formula: see text] [Formula: see text] where [Formula: see text] and [Formula: see text] is the Riesz potential, which appears as a nonlocal term in the equation. We investigate the nonexistence and existence of isolated singular solutions of Choquard equation under different range of the pair of exponent [Formula: see text]. Furthermore, we obtain qualitative properties for the minimal singular solutions of the equation.

Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity

2021 ◽  
Vol 19 (1) ◽  
pp. 259-267
Author(s):  
Liuyang Shao ◽  
Yingmin Wang
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solutions ◽  
Riesz Potential ◽  
Asymptotic Behavior Of Solutions ◽  
Suitable Assumption ◽  
Choquard Equation ◽  
Existence Of Positive Solutions

Abstract 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 .

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