DOI: http://dx.doi.org/10.18273/revint.v34n1-2016001
Lq estimates of functions in the kernel of
an elliptic operator and applications
GONZALO GARCÍA CAMACHO, LILIANA POSADA VERA*
Universidad del Valle, Departamento de Matemáticas, Cali, Colombia.
Abstract In this work, we will find a family of small functions ηy in the
Kernel of an operator defined in the intersection of the Sobolev space H2,q(Sn)
with the orthogonal complement in H1,2(Sn) of the first eigenspace of the
laplacian on Sn, parameterized with a variable y belonging to a small ball
contained in Bn+1. We will find Lq estimates of these functions and we
will use those estimates to find a subcritical solution to the scalar curvature
problem on Sn, and a solution
of a nonlinear elliptical problem related to that problem, where Fy1 : Sn → Sn
is a centered dilation.
Keywords: Sobolev spaces, conformal deformations, elliptic equations.
MSC2010: 53C21, 58J32, 46E35, 58E11.
Estimativos Lq de funciones en el núcleo de un
operador elíptico y aplicaciones
Resumen. En este trabajo, vamos a encontrar una familia de pequeñas funciones
ηy en el kernel de un operador definido en la intersección del espacio
de Sóbolev H2,q(Sn) con el complemento ortogonal en H1,2(Sn) del primer
espacio propio del laplaciano sobre Sn, parametrizado con una variable y
que pertenece a una pequeña bola contenida en Bn+1. Encontraremos estimativos
Lq de estas funciones, las cuales utilizaremos para encontrar una
solución subcrítica al problema de curvatura escalar sobre Sn y una solución
de un problema elíptico no lineal relacionado con este problema, donde Fy1 : Sn → Sn es una dilatación
centrada.
Palabras clave: Espacios de Sóbolev, deformaciones conformes, ecuaciones elípticas.
Texto Completo disponible en PDF
References
[1] Bahri A. and Coron J.M., "The scalar-curvature problem on the standard threedimensional sphere", J. Funct. Anal. 95 (1991), No. 1, 106-172.
[2] Chang Sun-Yung A., Gursky M.J. and Yang P.C., "The scalar curvature equation on the 2- and 3-sphere", Calc. Var. Partial Differential Equations 1 (1993), No. 2, 205-229.
[3] Escobar J.F. and García G., "Conformal metrics on the ball with zero scalar curvature and prescribed mean curvature on the boundary", J. Funct. Anal. 211 (2004), No. 1, 71-152.
[4] García G. and Posada V.L., "A priori estimates of the prescribed scalar curvature on the sphere", Revista de Ciencias 19 (2015), No. 1, 73-86.
[5] Han Z.C., "Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent", Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), No. 2, 159-174.
[6] Kazdan J. and Warner F., "Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvature", Ann. of Math. (2) 101 (1975), 317-331.
[7] Li Y.Y., "Prescribing scalar curvature on Sn and related problems. I", J. Differential Equations 120 (1995), No. 2, 319-410.
[8] Schoen R. and Zhang D., "Prescribed scalar curvature on the n-sphere", Calc. Var. Partial Differential Equations 4 (1996), No. 1, 1-25.
[9] Zhang D., "New results on geometric variational problems", Thesis (Ph.D), Stanford University, 1990, 85 p.
*E-mail: liliana.posada@correounivalle.edu.co
Received: 24 October 2015, Accepted: 21 January 2016.
To cite this article: G. García Camacho, L. Posada Vera, Lq estimates of functions in the kernel of an
elliptic operator and applications, Rev. Integr. Temas Mat. 34 (2016), No. 1, 1-21.