Electronic Journal of Differential Equations,
Vol. 1997(1997), No. 05, pp 1-11.

Title: On Neumann boundary value problems for some quasilinear elliptic 
       equations

Authors: Paul A. Binding (Univ. of Calgary, Canada)
         Pavel Drabek (Univ. of West Bohemia, Czech Republic)
         Yin Xi Huang (Univ. of Memphis, USA)

Abstract: We study the role played by the indefinite weight
function $a(x)$ on the existence of positive solutions to the problem
$$\left\{ \eqalign{
  -{\rm div}\,(|\nabla u|^{p-2}\nabla u)&=
  \lambda a(x)|u|^{p-2}u+b(x)|u|^{\gamma-2}u,
  \quad  x\in\Omega, \cr x{\partial u \over\partial n}&=0,  
  \quad x\in\partial\Omega\,,} \right. 
$$
where $\Omega$ is a smooth bounded domain in $R^n$, $b$ changes sign, 
$1<p<N$, $1<\gamma<Np/(N-p)$ and $\gamma\ne p$. 
We prove that 
(i) if $\int_\Omega a(x)\, dx\ne 0$ 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 $\int_\Omega b(x)\,dx<0$.

Submitted September 11, 1996. Published January 30, 1997.
Math Subject Classification: 35J65, 35J70, 35P30.
Key Words: p-Laplacian; positive solutions; Neumann boundary 
           value problems.