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Monotonicity for elliptic equations in unbounded Lipschitz domains. (English) Zbl 0906.35035
The authors investigate monotonicity properties for positive classical solutions $$u$$ of the boundary value problem $\Delta u+f(u)=0 \quad\text{in } \Omega, \quad u=0\quad\text{on }\Gamma: =\partial \Omega \tag{1}$ where $$\Omega$$ is an unbounded set defined as $$\Omega: =\{x\in\mathbb{R}^n |\;x_n> \varphi(x_1, \dots, x_{n-1})\}$$, with a globally Lipschitz continuous function $$\varphi: \mathbb{R}^{n-1} \to\mathbb{R}$$. Moreover $$u$$ is assumed to satisfy the condition $0<u< \sup u=M <\infty \quad \text{in } \Omega.\tag{2}$ The principal results of the paper are as follows: Theorem 1.1. Under the following conditions:
(a) $$f$$ is Lipschitz continuous on $$\mathbb{R}^+$$ and satisfies $$f(s)>0$$ on $$(0,\mu)$$ and $$f(s)\leq 0$$ for $$s\geq\mu$$ for some $$\mu>0$$; (b) for some $$0<s_0<s_1<\mu$$, $$f(s) >\delta_0s$$ on $$[0,s_0]$$ for some $$\delta_0 >0$$; (c) $$f(s)$$ is nonincreasing on $$(s_1,\mu)$$,
$$u$$ is monotonic with respect to $$x_n$$, i.e. $$\partial u/ \partial x_n>0$$ in $$\Omega$$.
Theorem 1.2. Under the assumptions of Theorem 1.1, the solution $$u$$ of (1) has in addition the following properties:
(a) $$u<\mu$$ in $$\Omega$$; (b) as $$\text{dist} (x,\Gamma) \to\infty$$, $$u(x) \to\mu$$ uniformly in $$\Omega$$; (c) $$u(x) \geq C[\text{dist} (x,\Gamma)]^\rho$$ if $$x_n-\varphi (x_1, \dots, x_{n-1}) <h_1$$ for some positive constants $$C,\rho_1$$ and $$h_1$$; (d) $$u$$ is the unique solution satisfying (1) and (2); (e) $$\partial u/ \partial x_n+ \sum^{n-1}_{\alpha=1} a_\alpha \partial u/ \partial x_\alpha>0$$ in $$\Omega$$ if $$\sum a^2_\alpha <\kappa^{-2}$$, where $$\kappa$$ is the Lipschitz constant of $$f$$.
The proofs of these results are established by use of the sliding method.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35B40 Asymptotic behavior of solutions to PDEs 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
##### Keywords:
uniqueness; positive classical solutions; sliding method
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