Let n be a positive integer with n≤8. Let u:Rn→R be a function. We say that u is a bounded classical solution to Δu+u−u3=0 if u is twice continuously differentiable, there exists a constant C such that ∣u(x)∣≤C for all x∈Rn, and Δu(x)+u(x)−u(x)3=0 for all x∈Rn. We say that u has positive first partial derivative if ∂1u(x)>0 for all x∈Rn. We say that u has hyperplane level sets if for every y in the range of u, there exists an affine subspace S⊆Rn of codimension 1 such that {x∈Rn:u(x)=y}=S. Prove that if u is a bounded classical solution to Δu+u−u3=0 with positive first partial derivative, then u has hyperplane level sets.