| Summary: | function $a(x)$ on the existence of positive solutions to the problem $$left{ eqalign{ -{ m div},(|abla u|^{p-2}abla u)&= lambda a(x)|u|^{p-2}u+b(x)|u|^{gamma-2}u, quad xinOmega, cr x{partial u overpartial n}&=0, quad xinpartialOmega,,} ight. $$ where $Omega$ is a smooth bounded domain in $R^n$, $b$ changes sign, $1<p<N$, $1<gamma<Np/(N-p)$ and $gammae p$. We prove that (i) if $int_Omega a(x), dxe 0$ and $b$ satisfies another integral condition, then there exists some $lambda^*$ such that $lambda^* int_Omega a(x), dx<0$ and, for $lambda$ strictly between 0 and $lambda^*$, the problem has a positive solution. (ii) if $int_Omega a(x), dx=0$, then the problem has a positive solution for small $lambda$ provided that $int_Omega b(x),dx<0$.
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