Sesión Ecuaciones Diferenciales y Probabilidad

Diciembre 14, 15:50 ~ 16:10

Nonradial nodal solutions to the Yamabe problem with maximal rank

Medina, María

In [2] M. Musso and Wei provided the first example in the literature of a nondegenerate nodal nonradial sign-changing solution (in the sense of Duyckaerts-Kenig-Merle [1]) to the equation \begin{equation}\label{prob} -\Delta u=\gamma |u|^{p-1}u\hbox{ in }\mathbb{R}^n,\qquad p=\frac{n+2}{n-2}, \qquad \gamma:=\frac{n(n-2)}{4},\qquad u\in\mathcal{D}^{1,2}(\mathbb{R}^n). \end{equation} In this work the authors left as an open question the existence of a solution of this type with maximal rank, i.e., a solution $u$ such that the kernel of the associated linearized operator $$L_u:=-\Delta -\gamma |u|^{p-2}u$$ has dimension exactly \begin{equation*} N:=2n+1+\frac{n(n-1)}{2}, \end{equation*} that is precisely the dimension of the family of transformations generated by the invariances of the equation (1) (traslation, dilation, rotation and Kelvin transform). By means of a Lyapunov-Schmidt reduction we will construct a nondegenerate solution as a perturbation of a sum of positive and negative rescaled copies of the ground state solution $$U(y):=\left(\frac{2}{1+|y|^2}\right)^{\frac{n-2}{2}},$$ appropriately located in the space. We will see that such solution is nondegenerate and has maximal rank whenever $n\geq 5$ and it is odd, giving a positive answer to the question formulated in [2] in those cases. This is a joint work with M. Musso and J. Wei. \begin{thebibliography}{9} \bibitem{DKM}{T. Duyckaerts, C. Kenig, F. Merle}, {\em Solutions of the focusing nonradial critical wave equation with the compactness property}, To appear in Ann. Sc. Norm. Super. Pisa Cl. Sci. \bibitem{MW}{M. Musso, J. Wei}, {\em Nondegeneracy of nodal solutions to the critical Yamabe problem.} Communications in Mathematical Physics, { Volume 340, Issue 3} (2015), 1049--1107. \end{thebibliography}

Autores: Medina, María / Musso, Monica / Wei, Juncheng.