On the Yamabe equation with rough potentials

Abstract

We study the existence of non--trivial solutions to the Yamabe equation: $$-\Delta u+ a(x)= \mu u|u|^\frac4{n-2} \hbox{} \mu >0, x\in \Omega \subset {\mathbf R}^n \hbox{with} n\geq 4,$$ $$ u(x)=0 \hbox{on} \partial \Omega$$ under weak regularity assumptions on the potential $a(x)$. More precisely in dimension $n\geq 5$ we assume that: \begin{enumerate} \item $a(x)$ belongs to the Lorentz space $L^{\frac n2, d}(\Omega)$ for some $1\leq d <\infty$, \item $a(x) \leq M<\infty \hbox{a.e.} x\in \Omega$, \item the set $\{x\in \Omega|a(x)<0\}$ has positive measure, \item there exists $c>0$ such that $$\int_\Omega (|\nabla u|^2 + a(x) |u|^2) \hbox{} dx \geq c\int_\Omega |\nabla u|^2 \hbox{} dx \hbox{} \forall u\in H^1_0(\Omega).$$ \end{enumerate} \noindent In dimension $n=4$ the hypothesis $(2)$ above is replaced by $$a(x)\leq 0 \hbox{} a.e. \hbox{} x\in \Omega.$$