Please use this identifier to cite or link to this item: http://hdl.handle.net/10077/4270
Title: On global solutions to a semilinear elliptic boundary problem in an unbounded domain
Authors: Egorov, Yuri V.
Kondratiev, Vladimir A.
Issue Date: 2000
Publisher: Università degli Studi di Trieste. Dipartimento di Scienze Matematiche
Source: Yury V. Egorov and Vladimir A. Kondratiev, "On global solutions to a semilinear elliptic boundary problem in an unbounded domain", in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 31 (2000) suppl.2, pp. 87-102.
Series/Report no.: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
31 (2000) suppl.2
Abstract: We consider solutions to the elliptic linear equation \[ Lu:=\underset{i,j=1}{\overset{n}{\sum}}\frac{\partial}{\partial x_{i}}\left(a_{ij}\left(x\right)\frac{\partial u}{\partial x_{j}}\right)=0\qquad\qquad\left(1\right) \] of second order in an unbounded domain \[ \left\{ x=\left(x',x_{n}\right)\::\:\mid x'\mid<Ax_{n}^{\sigma}+B,0<x_{n}<\infty\right\} ,0\leq\sigma\leq1, \] in $\mathbf{R}^{n}$. We study the asymptotic behiaviour as $x_{n}\rightarrow\infty$ of the solutions of $\left(1\right)$ satisfying the nonlinear boundary condition \[ \frac{\partial u}{\partial N}-b\left(x\right)\mid u\left(x\right)\mid^{p-1}u\left(x\right)=0\qquad\qquad\left(2\right) \] on the lateral surface \[ S=\left\{ x\epsilon\partial Q,\;0<x_{n}<\infty\right\} , \] where p>0, b(x)$\geq b_{0}$ >0. We show that a global solution of the problem can exist not for all values of parameters p, $\sigma$ and indicate these values. The boundary problem in the cylinder was studied by us in $\left[1\right]$,$\left[2\right]$. The obtained results generalize some results of B. Hu in $\left[4\right]$.
URI: http://hdl.handle.net/10077/4270
ISSN: 0049-4704
Appears in Collections:Rendiconti dell'Istituto di matematica dell'Università di Trieste: an International Journal of Mathematics vol.31 (2000) s2

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