scholarly journals On the asymptotic behavior of positive solutions of certain integral equations

2015 ◽  
Vol 44 ◽  
pp. 5-11
Author(s):  
Said R. Grace
2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Sufang Tang ◽  
Jingbo Dou

<p style='text-indent:20px;'>In this paper we analyzed the integrability and asymptotic behavior of the positive solutions to the Euler-Lagrange system associated with a class of weighted Hardy-Littlewood-Sobolev inequality on the upper half space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}_+^n. $\end{document}</tex-math></inline-formula> We first obtained the optimal integrability for the solutions by the regularity lifting theorem. And then, with this integrability, we investigated the growth rate of the solutions around the origin and the decay rate near infinity.</p>


2021 ◽  
Vol 19 (1) ◽  
pp. 259-267
Author(s):  
Liuyang Shao ◽  
Yingmin Wang

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 .


2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Habib Mâagli ◽  
Noureddine Mhadhebi ◽  
Noureddine Zeddini

We establish the existence and uniqueness of a positive solution for the fractional boundary value problem , with the condition , where , and is a nonnegative continuous function on that may be singular at or .


2015 ◽  
Vol 27 (3) ◽  
pp. 311-323
Author(s):  
Saïd Abbas ◽  
Mouffak Benchohra ◽  
Boualem A. Slimani ◽  
Juan J. Trujillo

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 180
Author(s):  
Oleg Avsyankin

The multidimensional integral equation of second kind with a homogeneous of degree (−n) kernel is considered. The special class of continuous functions with a given asymptotic behavior in the neighborhood of zero is defined. It is proved that, if the free term of the integral equation belongs to this class and the equation itself is solvable, then its solution also belongs to this class. To solve this problem, a special research technique is used. The above-mentioned technique is based on the decomposition of both the solution and the free term in spherical harmonics.


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