Multiplicity Result for a Degenerate Elliptic Equation with Logistic Reaction

A degenerate elliptic equation λΔpu+uq−1(1−ur)=0 with zero Dirichlet boundary condition, where λ is a positive parameter, 20, is studied. It has been known that there exists a positive number Λ such that if λ>Λ, then the problem has no positive solution; if λ Λ, then there exists a maximal solution; and for sufficiently small λ (<Λ), the problem has at least two solutions. In this article, it is shown that if λ<Λ, then there exist at least two positive solutions, via the variational method.