DOI: http://dx.doi.org/10.18273/revint.v34n1-2016001

L^q 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 η_y in the Kernel of an operator defined in the intersection of the Sobolev space H^2,q(Sⁿ) with the orthogonal complement in H^1,2(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 L_q 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 F_y1 : Sⁿ → Sⁿ is a centered dilation.

Keywords: Sobolev spaces, conformal deformations, elliptic equations.
MSC2010: 53C21, 58J32, 46E35, 58E11.

Estimativos L^q de funciones en el núcleo de un ]]> operador elíptico y aplicaciones

Resumen. En este trabajo, vamos a encontrar una familia de pequeñas funciones η_y en el kernel de un operador definido en la intersección del espacio de Sóbolev H^2,q(Sⁿ) con el complemento ortogonal en H^1,2(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 L^q 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 F_y1 : Sⁿ → Sⁿ es una dilatación centrada.

Palabras clave: Espacios de Sóbolev, deformaciones conformes, ecuaciones elípticas.

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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.         [ Links ]

[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.         [ Links ]

[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.         [ Links ]

[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.         [ Links ]

[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.         [ Links ]

[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.         [ Links ]

[7] Li Y.Y., "Prescribing scalar curvature on Sⁿ and related problems. I", J. Differential Equations 120 (1995), No. 2, 319-410.         [ Links ]

[8] Schoen R. and Zhang D., "Prescribed scalar curvature on the n-sphere", Calc. Var. Partial Differential Equations 4 (1996), No. 1, 1-25.         [ Links ]

[9] Zhang D., "New results on geometric variational problems", Thesis (Ph.D), Stanford University, 1990, 85 p.         [ Links ]

^*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, L^q estimates of functions in the kernel of an elliptic operator and applications, Rev. Integr. Temas Mat. 34 (2016), No. 1, 1-21.