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Zbl 0838.35009
Ni, Wei-Ming; Wei, Juncheng
On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems.
(English)
[J] Commun. Pure Appl. Math. 48, No.7, 731-768 (1995). ISSN 0010-3640

The authors consider problem (1): $\varepsilon^2\Delta u- u+ f(u)= 0$ and $u> 0$ in $\Omega$, $u= 0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in $\bbfR^n$, with smooth boundary $\partial\Omega$, and $f$ is a suitable function $\bbfR\to \bbfR$; the particular case $f(t)= t^p$, $1< p< (n+ 2)/(n- 2)$ is allowed. They state that, as $\varepsilon\to 0$, a least energy solution $u_\varepsilon$ to (1) has at most one local maximum which is achieved at exactly one point $P_\varepsilon\in \Omega$; furthermore $u_\varepsilon\to 0$ except at $P_\varepsilon$ and $d(P_\varepsilon, \partial\Omega)\to \max_{P\in \Omega} d(P, \partial\Omega)$, where $d$ denotes the distance function. Their approach is based on an asymptotic formula for the least positive critical value $c_\varepsilon$ of the energy $J_\varepsilon$ (i.e. $J_\varepsilon(u_\varepsilon)= c_\varepsilon$). In particular, they show that the dominating correction term in the expansion for $c_\varepsilon$, involves $d(P_\varepsilon, \partial\Omega)$ and is of order $\exp(- 1/\varepsilon)$. They make use of the vanishing viscosity method and methods developed earlier for the corresponding Neumann problem [the first author and {\it I. Takagi}, Commun. Pure Appl. Math. 44, No. 7, 819-851 (1991; Zbl 0754.35042) and Duke Math. J. 70, No. 2, 247-281 (1993; Zbl 0796.35056)].
[D.Huet (Nancy)]
MSC 2000:
*35B25 Singular perturbations (PDE)
35J60 Nonlinear elliptic equations
35B05 General behavior of solutions of PDE

Keywords: least energy solution; vanishing viscosity method

Citations: Zbl 0754.35042; Zbl 0796.35056

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