Monotone solutions to a class of elliptic and diffusion equations

In this paper, we study the existence and properties of monotone solutions to the following elliptic equation in $\mathbf R^n$ $-\Delta u= F'(u),$ in $\mathbf R^n,$ $\partial_{x_n}u>0,$ and the diffusion equation $u_t-\Delta u= F'(u),$ in $\mathbf R^n\times$ {$t>0$}, $\partial_{x_n}u>0, u|_{t=0}=u_0,$ where $\Delta$ is the standard Laplacian operator in $\mathbf R^n$, and $u_0$ is a given smooth function in $\mathbf R^n$ with some monotonicity condition. We show that under a natural condition on the nonlinear term $F'$, there exists a global solution to the diffusion problem above, and as time goes to infinity, the solution converges in $C_{l o c}^2(\mathbf R^n)$ to a solution to the corresponding elliptic problem. In particular, we show that for any two solutions $u_1(x')<$ $u_2(x')$ to the elliptic equation in $\mathbf R^{n-1}$: $-\Delta u=F'(u),$ in $\mathbf R^{n-1}, $ either for every $c\in (u_1(0),u_2(0))$, there exists an $(n-1)$ dimensional solution $u_c$ with $u_c(0)=c$, or there exists an $x_n$-monotone solution $u(x',x_n)$ to the elliptic equation in $\mathbf R^n$: $-\Delta u=F'(u), $ in $\mathbf R^n,$ $\partial_{x_n}u>0,$ in $\mathbf R^n$ such that $\lim_{x_n\to-\infty}u(x',x_n)=v_1(x')\leq u_1(x')$ and $\lim_{x_n\to+\infty}u(x',x_n)=v_2(x')\leq u_2(x').$ A typical example is when $F'(u)=u-|u|^{p-1}u$ with $p>1$. Some of our results are similar to results for minimizers obtained by Jerison and Monneau [13] by variational arguments. The novelty of our paper is that we only assume the condition for $F$ used by Keller and Osserman for boundary blow up solutions.