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We consider the nonlinear eigenvalue problem $$-{div\left(\right|\nabla u|}^{p-2}{\nabla u)=\lambda g\left(x\right)|u|}^{p-2}u$$ in ${𝐑}^N$ 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 $W^{1,p}\left({𝐑}^N\right)$. A nonexistence result is also given for the case $p\ge N$.

We consider the existence of positive solutions of -pu=g(x)|u|p-2u+h(x)|u|q-2u+f(x)|u|p*-2u(1) in ${ℝ}^N$, where $\lambda ,\alpha \in ℝ$, $1<p<N$, $p^{*}=Np/(N-p)$, the critical Sobolev exponent, and $1<q<p^{*}$, $q\ne p$. Let ${\lambda }_1^{+}>0$ be the principal eigenvalue of -pu=g(x)|u|p-2u    in ,        g(x)|u|p>0, (2) with $u_1^{+}>0$ the associated eigenfunction. We prove that, if ${\int }_{{ℝ}^N}f{\left|u_1^{+}\right|}^{p^{*}}<0$, ${\int }_{{ℝ}^N}h{\left|u_1^{+}\right|}^q>0$ if $1<q<p$ and ${\int }_{{ℝ}^N}h{\left|u_1^{+}\right|}^q<0$ if $p<q<p^{*}$, then there exist ${\lambda }^{*}>{\lambda }_1^{+}$ and ${\alpha }^{*}>0$, such that for $\lambda \in [{\lambda }_1^{+},{\lambda }^{*})$ and $\alpha \in [0,{\alpha }^{*})$, (1) has at least one positive solution.