Rigidity of solutions to elliptic equations with one uniform limit

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