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

• Main Authors: Gustavo Izquierdo B., Gabriel Lopez G.
• Format: Article
• Language: English
• Published: Texas State University 2005-10-01
• Series: Electronic Journal of Differential Equations
• Subjects: Resonance; p-Laplacian; improved Poincare inequality.
• Online Access: http://ejde.math.txstate.edu/Volumes/2005/112/abstr.html

We show the existence of a weak solution for the problem $$ -Delta_p u=lambda_1h(x)|u|^{p-2}u+a(x)g(u)+f(x),quad uinmathcal{D}^{1,p}(mathbb{R}^N), $$ 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 $sinmathbb{R}$, $ain L^{(p^{*})'}(mathbb{R}^N)cap L^{infty}(mathbb{R}^N)$ and $fin L^{(p^{*})'}(mathbb{R}^N)$. 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].