On the p-laplacian with weights



Cecilia Yarur
Universidad de Santiago de Chile
Departamento de Matematicá
email: cyarur@fermat.usach.cl



Abstract: Let $B,C:(0,\infty)\mapsto (0,\infty)$ be two given weight functions and consider the equation

\begin{displaymath} -\hbox{div}(\vert\nabla u\vert^{p-2}\nabla u)=\lambda C(\ver... ...t^{p-2}u+B(\vert x\vert)\vert u\vert^{q-2}u,\quad x\in B_1(0), \end{displaymath}

where $q>p>1$. By considering positive radial bounded solutions to this equation with zero at the boundary, we are led to study the problem

\begin{displaymath} \begin{cases}-(r^{N-1}\vert u'\vert^{p-2}u')'=\lambda c(r)\v... ...r\to 0}r^{N-1}\vert u'(r)\vert^{p-1}=0,\quad u(1)=0 \end{cases}\end{displaymath}

where $b(r)=r^{(N-1)}B(r)$ and $c(r)=r^{(N-1)}C(r)$. We study Brezis-Nirenberg type problems for this equation.