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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^*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.