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On global solutions to a semilinear elliptic boundary problem in an unbounded domain. (English) Zbl 0965.35049

The authors investigate the solutions of the elliptic linear equation \[ \sum\limits_{i,j=1}^n\partial /\partial x_i(a_{ij}(x)\partial u/\partial x_j)=0 \] in an unbounded domain \(\{x=(x',x_n):|x'|<Ax_n^{\sigma }+B\), \(0<x_n<\infty \}\), \(0\leq \sigma \leq 1\), in \(\mathbb{R}^n\). It is considered a problem satisfying the nonlinear boundary condition \(\partial u/\partial N-b(x)|u(x)|^{p-1}u(x)=0\) on the lateral surface \(S=\{x\in \partial Q, \;0<x_n<\infty \}\), where \(p>0\), \(b(x)\geq b_0>0\). The asymptotic behavior as \(x_n\to \infty \) of the solutions to the considered problem is studied. In addition, it is shown that a global solution of the problem can exist not for all values of the parameters \(p\), \(\sigma \). The obtained results are a generalization of the results of B. Hu [Differ. Integral Equ. 7, No. 2, 301-313 (1994; Zbl 0820.35062)].

MSC:

35J60 Nonlinear elliptic equations
35J45 Systems of elliptic equations, general (MSC2000)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)

Citations:

Zbl 0820.35062
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