Let n=7. Consider a function u:Rn→R. 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 everywhere positive first partial derivative if the directional derivative of u in the direction of the first coordinate vector is strictly positive at every point. We say that u has hyperplane level sets if for every value y in the range of u, there exists an affine subspace S⊂Rn of codimension 1 such that the level set {x∈Rn:u(x)=y} equals S. Prove that if u is a bounded classical solution to Δu+u−u3=0 with everywhere positive first partial derivative, then u has hyperplane level sets.