Kelvin transform for Grushin operators and critical semilinear equations
2005; Duke University Press; Volume: 131; Issue: 1 Linguagem: Inglês
10.1215/s0012-7094-05-13115-5
ISSN1547-7398
AutoresRoberto Monti, Daniele Morbidelli,
Tópico(s)Geometric Analysis and Curvature Flows
ResumoWe study positive entire solutions u=u(x,y) of the critical equation Δxu+(α+1)2|x|2αΔyu=-u(Q+2)/(Q-2) inℝn=ℝm×ℝk, where (x,y)∈ℝm×ℝk, α>0, and Q=m+k(α+1). In the first part of the article, exploiting the invariance of the equation with respect to a suitable conformal inversion, we prove a "spherical symmetry result for solutions". In the second part, we show how to reduce the dimension of the problem using a hyperbolic symmetry argument. Given any positive solution u of (1), after a suitable scaling and a translation in the variable y, the function v(x)=u(x,0) satisfies the equation divx(p∇xv)-qv=-pv(Q+2)/(Q-2), |x|<1, with a mixed boundary condition. Here, p and q are appropriate radial functions. In the last part, we prove that if m=k=1, the solution of (2) is unique and that for m≥3 and k=1, problem (2) has a unique solution in the class of x-radial functions
Referência(s)