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Singularly perturbed semilinear elliptic equations in unbounded domains. (Chinese. English summary) Zbl 0816.35028

The author considers a boundary value problem for the following semilinear elliptic equation with singular perturbation in an unbounded domain: \[ L_ 2(u)+ \varepsilon L_ 1(u)= f(x,u), \qquad x\in\Omega, \] where \(L_ 2(u)\) is a strictly elliptic operator, \(L_ 1(u)\) is a linear operator, \(\varepsilon>0\) is a perturbation parameter, and \(\Omega\subset \mathbb{R}^ n\) is unbounded. The author proves by means of a differential inequality method under certain conditions that there exists a solution to the above problem and that this solution has an asymptotic expansion, which is uniformly valid in the whole domain.

MSC:

35J60 Nonlinear elliptic equations
35C20 Asymptotic expansions of solutions to PDEs
35B25 Singular perturbations in context of PDEs
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