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Inequalities for second-order elliptic equations with applications to unbounded domains. I. (English) Zbl 0860.35004
In recent papers, the authors have studied symmetry and monotonicity properties for positive solutions $$u$$ of elliptic equations of the form $u>0,\;\Delta u+f(u)=0 \text{ in } \Omega, \quad u=0 \text{ on } \partial\Omega \tag{1}$ in several classes of unbounded domains $$\Omega$$ in $$\mathbb{R}^n$$. Here they continue this program by considering another type of domain, $$\Omega= \mathbb{R}^{n-j} \times\omega$$, where $$\omega$$ is a smooth bounded domain in $$\mathbb{R}^j$$.
Denote by $$x=(x_1, \dots, x_{n-j})$$ the coordinates in $$\mathbb{R}^{n-j}$$, and by $$y=(y_1, \dots, y_j)$$ the coordinates in $$\omega$$. The goal is to establish symmetry of solutions of (1) corresponding to symmetries of $$\omega$$. For example, if $$\omega$$ is a ball $$\{|y |<R\}$$, they prove that any solution of (1) depends only on $$|y|$$ and $$x$$, and is decreasing in $$|y|$$. Note that $$u$$ is not assumed to be bounded. Throughout the paper it is assumed that $$f$$ is Lipschitz continuous, with Lipschitz constant $$k$$, on $$\mathbb{R}^+$$ (or on $$[0, \sup u]$$ in the case where $$u$$ is bounded).
Reviewer: V.Mustonen (Oulu)

##### MSC:
 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations 35B40 Asymptotic behavior of solutions to PDEs
##### Keywords:
semilinear elliptic equation; symmetry of solutions
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##### References:
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