Let n=5. 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, where Δu denotes the Laplacian of u. We say that u has positive first partial derivative if ∂1u(x)>0 for all x∈Rn, meaning the directional derivative of u in the direction of the first standard basis vector is strictly positive everywhere. We say that u has hyperplane level sets if for every y in the range of u, there exists an affine subspace S⊂Rn such that the level set u−1({y})={x∈Rn:u(x)=y} equals S and the direction of S has dimension n−1=4. 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.