Existence and regularity of solutions for a semilinear first-order equation on the torus

Abstract

The problem $$\sum {a_i \frac{{\partial u}}{{\partial x_i }} + g(x,u) = 0}$$ is considered on the N-dimensional torus Ω with ai∈ℝ and g a continuous function satisfying a growth condition as ¦u¦→∞. We show the existence of bounded solutions that are continuous if g is strictly increasing in u.