Abstract:We consider the nonlinear eigenvalue problem $$ -div(|{\nabla }u|^{p-2}{\nabla }u)={\lambda }g(x)|u|^{p-2}u $$ in $\boldkey R^N$ with $p>1$. A condition on indefinite weight function $g$ is given so that the problem has a sequence of eigenvalues tending to infinity with decaying eigenfunctions in ${W^{1, p}(\boldkey R^N)}$. A nonexistence result is also given for the case $p\geq N$.
Keywords: eigenvalue, the $p$-Laplacian, indefinite weight, $\boldkey R^N$
AMS Subject Classification: Primary 35P30, 35J70