On rapidly oscillating solutions of a nonlinear elliptic equation. (English) Zbl 1481.35025
Summary: The aim of this article is to examine the solutions of the boundary value problem of the nonlinear elliptic equation \(\varepsilon^2\Delta u=f(u)\). We describe the asymptotic behavior as \(\varepsilon\) tends to zero of the solutions on a spherical crown \(C\) of \(\mathbb R^N\), \((N\geq 2)\) in a direct non-classical formulation which suggests easy proofs. We propose to look for interesting solutions in the case where the condition at the edge of the crown is a constant function. Our results are formulated in classical mathematics. Their proofs use the stroboscopic method which is a tool of the nonstandard asymptotic theory of differential equations.
MSC:
| 35B25 | Singular perturbations in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35J61 | Semilinear elliptic equations |
Keywords:
non standard analysis (NSA); reduced problem; vector field; stroboscopic method; solutions on a spherical crownReferences:
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