Classification of solutions for an integral equation. (English) Zbl 1093.45001

Chen, Wenxiong; Li, Congming; Ou, Biao

Commun. Pure Appl. Math. 59, No. 3, 330-343 (2006); corrigendum 59, No. 7, 1064 (2006).

The authors consider the integral equation \[ u\left( x\right) =\int_{{\mathbb R}^n}\frac 1{\left| x-y\right| ^{n-\alpha }}u\left( y\right) ^{\frac{n+\alpha }{n-\alpha }}dy \] which is equivalent to the semilinear partial differential equation \[ \left( -\Delta \right) ^{\alpha /2}u=u^{\frac{n+\alpha }{n-\alpha }}. \] Using the method of moving planes in an integral form, the authors prove that every positive regular solution \(u\left( x\right) \) is radially symmetric and monotone about some point and therefore one shows that the only possible form of the solution that one can assume is the form \( \displaystyle c\left( \frac t{t^2+\left| x-x_0\right| ^2}\right) ^{\frac{ n-\alpha }2}.\)

Reviewer: Adrian Carabineanu (Bucureşti)

MSC:

45E10	Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
45M20	Positive solutions of integral equations
35J65	Nonlinear boundary value problems for linear elliptic equations

Keywords:

Integral equation; moving planes method; semilinear partial differential equations; radially symmetric solution; monotone solution; positive regular solution

References:

[1]	Caffarelli, Comm Pure Appl Math 42 pp 271– (1989)
[2]	Chang, Math Res Lett 4 pp 91– (1997) · Zbl 0903.53027 · doi:10.4310/MRL.1997.v4.n1.a9
[3]	Chen, Duke Math J 63 pp 615– (1991)
[4]	Chen, Ann of Math (2) 145 pp 547– (1997)
[5]	Chen, Commun Pure Appl Anal 4 pp 1– (2005)
[6]	Chen, Comm Partial Differential Equations 30 pp 59– (2005)
[7]	Chen, Discrete Contin Dyn Syst 12 pp 347– (2005)
[8]	An introduction to maximum principles and symmetry in elliptic problems. Cambridge Tracts in Mathematics, 128. Cambridge University Press, Cambridge, 2000. · Zbl 0947.35002 · doi:10.1017/CBO9780511569203
[9]	; ; Symmetry of positive solutions of nonlinear elliptic equations in Rn. Mathematical analysis and applications, Part A, 369–402. Advances in Mathematics, Supplementary Studies, 7a. Academic Press, New York-London, 1981.
[10]	Li, Invent Math 123 pp 221– (1996) · Zbl 0849.35009 · doi:10.1007/s002220050023
[11]	Li, J Eur Math Soc (JEMS) 6 pp 153– (2004)
[12]	Lieb, Ann of Math 118 pp 349– (1983)
[13]	; Analysis. 2nd edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, R.I., 2001. · doi:10.1090/gsm/014
[14]	Ou, Houston J Math 25 pp 181– (1999)
[15]	Serrin, Arch Rational Mech Anal 43 pp 304– (1971)
[16]	Wei, Math Ann 313 pp 207– (1999)