Non-local elliptic problem in higher dimension
Tosiya Miyasita · Osaka City University (Osaka City University) · 2007
Non-local elliptic problem, $-\\Delta v = \\lambda \\bigl(e^{v}\\big/\\bigl(\\int_{\\Omega}e^{v} dx \\bigr)^{p}\\bigr)$ with Dirichlet boundary condition is considered on $n$-dimensional bounded domain $\\Omega$ with $n \\geq 3$ for $p>0$. If $\\Omega$ is the unit ball, $3 \\leq n \\leq 9$ and $2/n \\leq p \\leq 1$, we have infinitely many bendings in $\\lambda$ of the solution set in $\\lambda-v$ plane. Finally if $\\Omega$ is an annulus domain and $p \\geq 1$, we show that a solution exists for all $\\lambda>0$.