Existence and multiplicity results for generalized logistic equations

We consider elliptic Dirichlet problem −Δpu=λf(x,u,∇u)−g(x,u)  in  Ω,u=0  on  ∂Ω−Δpu=λf(x,u,∇u)−g(x,u)  in  Ω,u=0  on  ∂Ω. Assume that the nonlinearity ff 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.