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positive solutions; critical exponent; the $p$-Laplacian
We consider the existence of positive solutions of -\Delta_pu=\lambda g(x)|u|^{p-2}u+\alpha h(x)|u|^{q-2}u+f(x)|u|^{p^*-2}u\eqno(1) in $\Bbb R^N$, where $\lambda, \alpha\in\Bbb R$, $1<p<N$, $p^*=Np/(N-p)$, the critical Sobolev exponent, and $1<q<p^*$, $q\ne p$. Let $\lambda_1^+>0$ be the principal eigenvalue of -\Delta_pu=\lambda g(x)|u|^{p-2}u \quad\text{in} \Rn, \qquad\int_{\Rn} g(x)|u|^p>0, \eqno(2) with $u_1^+>0$ the associated eigenfunction. We prove that, if $\int_{\Bbb R^N}f|u_1^+|^{p^*}<0$, $\int_{\Bbb R^N}h|u_1^+|^q>0$ if $1<q<p$ and $\int_{\Bbb R^N}h|u_1^+|^q<0$ 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.
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