Dirichlet problems for the p-Laplacian with a convection term

We consider the nonlinear Dirichlet boundary value problem in a bounded domain Ω ⊂ R N with smooth boundary ∂ Ω , where Δ p u = def div ( | ∇ u | p - 2 ∇ u ) with 1 < p < ∞ , λ ∈ R , and h ∈ L ∞ ( Ω ) . The term B ( x , ∇ u ) is a continuous function assumed to be also homogeneous of degree ( p - 1 ) and odd with respect to the second variable; B ( x , η ) = ( a ( x ) · η ) | η | p - 2 being a canonical example with a given vector field a ∈ [ C ( Ω ¯ ) ] N , for ( x , η ) ∈ Ω × R N . For the corresponding eigenvalue problem obtained by setting h ≡ 0 , we show existence, simplicity, and isolation of the principal eigenvalue λ 1 ( λ 1 > 0 ). When h ≢ 0 and - ∞ < λ < λ 1 , we prove that there exists a weak solution u ∈ W 0 1 , p ( Ω ) to problem ( P ); this solution is unique provided λ < 0 (without any further assumptions). When h ≥ 0 , h ≢ 0 , and 0 ≤ λ < λ 1 , we show that the solution is positive and also unique.