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A higher-order energy expansion to two-dimensional singularly perturbed Neumann problems. (English) Zbl 1083.35038

The authors deal with the following singularly perturbed semilinear elliptic problem \[ \varepsilon^2\Delta u- u+ u^p= 0\quad\text{in }\Omega, \]
\[ u> 0\quad\text{in }\Omega, \]
\[ {\partial u\over\partial n}= 0\quad\text{on }\partial\Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb R^N\) with smooth boundary \(\partial\Omega\), \(\varepsilon> 0\) is a small constant and \(1< p<({N+2\over N-2})_+\). Since, in the two-dimensional case, the scalar curvature is zero, the well-known asymptotic expansions of the first two authors [C. R., Math., Acad. Sci. Paris 337, No. 1, 37–42 (2003; Zbl 1084.35023), Calc. Var. Partial Differ. Equ. 20, No. 4, 403–430 (2004; Zbl 1154.35353)] for the corresponding energy functional is not enough to locate the spike if there are several maximum points of the mean curvature, so that the next order (higher) term in asymptotic expansions of the energy is required. The authors discuss this issue.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B25 Singular perturbations in context of PDEs
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J60 Nonlinear elliptic equations
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