Radial Symmetry and Asymptotic Estimates for Positive Solutions to a Singular Integral Equation

• Published in: Taiwanese journal of mathematics Vol. 20; no. 2; pp. 473 - 489
• Main Author: Lei, Yutian
• Format: Journal Article
• Language: English
• Published: Mathematical Society of the Republic of China 01.04.2016
• Subjects: Differential equations; Mathematical constants; Mathematical functions; Singular integral equations
• ISSN: 1027-5487; 2224-6851
• DOI: 10.11650/tjm.20.2016.6150

In this paper, we are concerned with the nonlinear singular integral equation u ( x ) = | x | σ ∫ R n u p ( y ) d y | x − y | n − α , where α ∈ ( 0 , n ) , σ ∈ ( max { − α , α − n 2 } , 0 ] . Such an integral equation appears in the study of sharp constants of the Hardy-Sobolev inequality and the Hardy-Littlewood-Sobolev inequality. It is often used to describe the shapes of the extremal functions. If 0 < p ≤ n n − α − σ , there is not any positive solution to this equation. Under the assumption of p = n + α + 2 σ n − α , we obtain an integrability result for the integrable solutionu(i.e., u ∈ L 2 n n − α ( ℝ n ) ) of the integral equation. Such an integrable solution is radially symmetric and decreasing aboutx 0∈ ℝ n . Furthermore,x 0is also the origin ifσ≠ 0. In addition, this integrable solution is blowing up with the rate −σwhen |x| → 0. Moreover, ifn+pσ> 0, thenudecays fast with the raten−α−σwhen |x| → ∞. 2010Mathematics Subject Classification. 45E10, 45G05. Key words and phrases. Riesz potential, Singular integral equation, Decay rates, Integrable solution, Bounded decaying solution.