Electron. J. Diff. Eqns. Vol. 1997(1997), No. 05, pp 1-11.

On Neumann boundary value problems for some quasilinear elliptic equations

Paul A. Binding, Pavel Drabek, & Yin Xi Huang

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 less than p less than N , 1 less than $\gamma$ less than Np/(N-p) and $\gamma\ne p$ . We prove that

  1. if $\int_\Omega a(x)\, dx\ne 0$ and b satisfies another integral condition, then there exists some $\lambda^*$ such that $\lambda^* \int_\Omega a(x)\, dx$ is negative and, for $\lambda$ strictly between 0 and $\lambda^*$ , the problem has a positive solution.
  2. if $\int_\Omega a(x)\, dx=0$ , then the problem has a positive solution for small $\lambda$ provided that $\int_\Omega b(x)\,dx$ is negative .

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

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Paul A. Binding
Department of Mathematics & Statistics, University of Calgary, Calgary, Alberta, Canada, T2N 1N4
e-mail: binding@acs.ucalgary.ca
photo Pavel Drabek
Department of Mathematics
University of West Bohemia
P.O. Box 314, 30614 Pilsen, Czech Republic
e-mail: pdrabek@kma.zcu.cz
Yin Xi Huang
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152 USA
e-mail: huangy@mathsci.msci.memphis.edu
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