Fibered nonlinearities for p(x)-Laplace equations

The purpose of this paper is to give some geometric results on the following problem: −div ( α(x)|∇u(X)|p(x)−2∇u(X) ) = f(x, u(X)) in Ω, (1.1) where f = f(x, u) ∈ L∞(Rm×R) is differentiable in u with fu ∈ L∞(R), α ∈ L∞(Rm), with inf Rm α > 0, p ∈ L∞(Rm), with p(x) ≥ 2 for any x ∈ R, and Ω is an open subset of R. Here, u = u(X), with X = (x, y) ∈ R × Rn−m. As well known, the operator in (1.1) comprises, as main example, the degenerate p(x)-Laplacian (and, in particular, the degenerate p-Laplacian). The motivation of this paper is the following. In [15], it was asked whether or not the level sets of bounded, monotone, global solutions of −∆u(X) = u(X) − u(X) (1.2) for X ∈ R, are flat hyperplanes, at least when n ≤ 8. In spite of the marvelous progress performed in this direction (see, in particular, [43, 8, 31, 32, 7, 5, 46, 16]), part of the conjecture and many related problems are still unsolved (see [27]). In [47], the following generalization of (1.2) was taken into account: −∆u(X) = f(x, u(X)). (1.3) where, as above, the notation X = (x, y) ∈ R × Rn−m is used.