Uniqueness for Neumann Problems for Nonlinear Elliptic Equations With Lower Order Terms

In this paper, we prove uniqueness results for weak solutions to a class of Neumann problems, whose prototype is ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝜆(1 + 𝑢2 ) ( 𝑝− 2)∕2 𝑢 −div ((1 + |∇ 𝑢|2 ) ( 𝑝− 2)∕2 ∇ 𝑢) −div ( 𝑐( 𝑥)(1 + |𝑢|2 ) ( 𝜏+ 1)∕2 ) + 𝑏( 𝑥)(1 + |∇ 𝑢|2 ) ( 𝜎+ 1)∕2 = 𝑓 in Ω( (1 + |∇ 𝑢|2 ) ( 𝑝− 2)∕2 ∇ 𝑢 + 𝑐( 𝑥)(1 + |𝑢|2 ) ( 𝜏+ 1)∕2 )) ⋅𝑛 = 0 on 𝜕Ω, where Ωis a bounded open subset of ℝ𝑁 ( 𝑁 ≥ 2) with Lipschitz boundary, 𝑝is a real number 2 𝑁 𝑁+ 1 < 𝑝 < 𝑁, the coefficients 𝑐( 𝑥) and 𝑏( 𝑥) belong to suitable Lebesgue spaces and 𝑓is an element of the dual space of the Sobolev space 𝑊1 ,𝑝 having a suitable summability. Finally, 𝜏and𝜎arepositiveconstantswhich belongtosuitableintervals specified in Theorems 2.3, 2.6, and 2.8. Uniqueness results for weak solutions are proved under smallness assumptions on the coefficients 𝑏 or 𝑐.