Existence and multiplicity results for generalized logistic equations

Abstract

We consider elliptic Dirichlet problem −Δрu=λf(x,u,∇u)−g(x,u) in Ω,u=0 on ∂Ω. Assume that the nonlinearity f satisfies certain growth condition. Using the fixed point index, the Leggett–Williams theorem on fixed point and the arguments on monotone minorant, we prove the existence and multiplicity results for the equation. This extends some known results.