Electronic Journal of Differential Equations (May 2004)
Variational methods for a resonant problem with the p-Laplacian in $R^N$
Abstract
The solvability of the resonant Cauchy problem $$ - Delta_p u = lambda_1 m(|x|) |u|^{p-2} u + f(x) quadhbox{in } mathbb{R}^N ;quad uin D^{1,p}(mathbb{R}^N), $$ in the entire Euclidean space $mathbb{R}^N$ ($Ngeq 1$) is investigated as a part of the Fredholm alternative at the first (smallest) eigenvalue $lambda_1$ of the positive $p$-Laplacian $-Delta_p$ on $D^{1,p}(mathbb{R}^N)$ relative to the weight $m(|x|)$. Here, $Delta_p$ stands for the $p$-Laplacian, $mcolon mathbb{R}_+o mathbb{R}_+$ is a weight function assumed to be radially symmetric, $m otequiv 0$ in $mathbb{R}_+$, and $fcolon mathbb{R}^No mathbb{R}$ is a given function satisfying a suitable integrability condition. The weight $m(r)$ is assumed to be bounded and to decay fast enough as $ro +infty$. Let $varphi_1$ denote the (positive) eigenfunction associated with the (simple) eigenvalue $lambda_1$ of $-Delta_p$. If $int_{mathbb{R}^N} fvarphi_1 ,{ m d}x =0$, we show that problem has at least one solution $u$ in the completion $D^{1,p}(mathbb{R}^N)$ of $C_{mathrm{c}}^1(mathbb{R}^N)$ endowed with the norm $(int_{mathbb{R}^N} | abla u|^p ,{ m d}x)^{1/p}$. To establish this existence result, we employ a saddle point method if $1 < p < 2$, and an improved Poincar'e inequality if $2leq p< N$. We use weighted Lebesgue and Sobolev spaces with weights depending on $varphi_1$. The asymptotic behavior of $varphi_1(x)= varphi_1(|x|)$ as $|x|o infty$ plays a crucial role. end{abstract}
