On the Yamabe equation with rough potentials

Abstract

We study the existence of non--trivial solutions to the Yamabe equation: $$-\Delta u+a(x)=\mu u|u{|}^{\frac{4}{n-2}}\mu >0,x\in \Omega \subset {\mathbf{R}}^{n}withn\ge 4,$$ $$u(x)=0on\partial \Omega$$ under weak regularity assumptions on the potential $a(x)$. More precisely in dimension $n\ge 5$ we assume that: \begin{enumerate} \item $a(x)$ belongs to the Lorentz space $L^{\frac{n}{2},d}(\Omega )$ for some $1\le d<\infty$, \item $a(x)\le M<\infty a.e.x\in \Omega$, \item the set $\{x\in \Omega |a(x)<0\}$ has positive measure, \item there exists $c>0$ such that $${\int }_{\Omega }(|\nabla u{|}^2+a(x)|u{|}^2)dx\ge c{\int }_{\Omega }|\nabla u{|}^{2}dx\forall u\in H_0^1(\Omega ).$$ \end{enumerate} \noindent In dimension $n=4$ the hypothesis $(2)$ above is replaced by $$a(x)\le 0a.e.x\in \Omega .$$