Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 112, pp. 1-8.

Title: A resonance problem for the p-laplacian in $\mathbb{R}^N$

Authors: Gustavo Izquierdo B. (Univ. Autonoma Metropolitana, Mexico)
         Gabriel Lopez G. (Univ. Autonoma Metropolitana, Mexico)

Abstract: 
 We show the existence of a weak solution for the problem
 $$
 -\Delta_p u=\lambda_1h(x)|u|^{p-2}u+a(x)g(u)+f(x),\quad
 u\in\mathcal{D}^{1,p}(\mathbb{R}^N),
 $$
 where, $2<p<N$, $\lambda_1$ is the first eigenvalue of the
 $p$-Laplacian on $\mathcal{D}^{1,p}(\mathbb{R}^N)$ relative to
 the radially symmetric weight $h(x)=h(|x|)$. In this problem,
 $g(s)$ is a bounded function for all $s\in\mathbb{R}$,
 $a\in L^{(p^{*})'}(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N)$ and
 $f\in L^{(p^{*})'}(\mathbb{R}^N)$. To establish an existence result,
 we employ the Saddle Point Theorem of Rabinowitz [9] and an
 improved Poincare inequality from an  article of Alziary,
 Fleckinger and Takac [2].
  
Submitted May 31, 2005. Published October 17, 2005.
Math Subject Classifications: 35J20.
Key Words: Resonance; p-Laplacian; improved Poincare inequality