Rigidity of solutions to elliptic equations with one uniform limit

Let \(u\ge -1\) be a solution to the semilinear elliptic equation \(-\Delta u = f(u)\) in \(\mathbb {R}^N\) such that \(\lim _{x_N\rightarrow -\infty } u(x',x_N) = -1\) uniformly in \(x'\in \mathbb {R}^{N-1}\), \(\lim _{t\rightarrow +\infty } \inf _{x_N>t} u(x) > -1\), and u is bounded in each half-space \(\{x_N<\lambda \}\), \(\lambda \in \mathbb {R}\). Here \(f:[-1,+\infty )\rightarrow \mathbb {R}\) is a locally Lipschitz continuous function which satisfies some mild assumptions. We show that u is strictly monotonically increasing in the \(x_N\)-direction. Under some further assumptions on f, we deduce that u depends only on \(x_N\) and it is unique up to a translation. In particular, such a solution u to the problem \(\Delta u = u + 1\) in \(\mathbb {R}^N\) must have the form \(u(x)\equiv e^{x_N+\alpha }-1\) for some \(\alpha \in \mathbb {R}\).