Nonradial positive solutions of the -Laplace Emden-Fowler equation with sign-changing weight

摘要

In this paper we study the p-Laplace Emden-Fowler equation with a radial and sign-changing weight in the unit ball under the Dirichlet boundary condition. We show that if the weight function is negative in the unit ball except for a small neighborhood of the boundary and positive at somewhere in this neighborhood, then no least energy solution is radially symmetric. Therefore the equation has both a positive radial solution and a positive nonradial solution. Moreover, we prove in the one dimensional case that if the neighborhood is large, then a positive solution is unique.