Electron. J. Diff. Eqns., Vol. 2005(2005), No. 112, pp. 1-8.

A resonance problem for the p-laplacian in $R^N$

Gustavo Izquierdo B., Gabriel Lopez G.

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 les than p less than 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

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Gustavo Izquierdo Buenrostro
Dept. Mat. Universidad Autonoma Metropolitana
Mexico
email: iubg@xanum.uam.mx
Gabriel Lopez Garza
Dept. Mat. Universidad Autonoma Metropolitana
Mexico
email: grlzgz@xanum.uam.mx

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