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eigenvalue; the $p$-Laplacian; indefinite weight; $\boldkey R^N$
We consider the nonlinear eigenvalue problem $$ -\operatorname{div}(|{\nabla} u|^{p-2}{\nabla} u)=\lambda g(x)|u|^{p-2}u $$ 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 $p\geq N$.
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