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.


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
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