L^q estimates of functions in the kernel of an elliptic operator and applications

Abstract

In this work, we will find a family of small functions $\eta_{y}$ in the Kernel of an operator defined in the intersection of the Sobolev space $H^{2,q}(S^{n})$ with the orthogonal complement in $H^{1,2}(S^{n})$ of the first eigenspace of the laplacian on $S^{n}$, parameterized with a variable $y$ belonging to a small ball contained in $B^{n+1}$. 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 $S^n$, and a solution $u_{y_{1}}=\alpha_{F_{y_{1}}^{-1}}(1+\eta_{y_{1}})=|F_{y_{1}}'|^{\frac{n-2}{2}}(1+\eta_{y_{1}})\circ F_{y_{1}}$ of a nonlinear elliptical problem related to that problem, where $F_{y_{1}}:S^{n}\rightarrow S^{n}$ is a centered dilation.

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.