On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems. (English) Zbl 0838.35009

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 \(\mathbb{R}^n\), with smooth boundary \(\partial\Omega\), and \(f\) is a suitable function \(\mathbb{R}\to \mathbb{R}\); 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 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)].
Reviewer: D.Huet (Nancy)


35B25 Singular perturbations in context of PDEs
35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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[1] Sobolev Spaces, Academic Press, New York, 1975. · Zbl 0314.46030
[2] Amann, Indiana Univ. Math. J. 27 pp 779– (1978)
[3] Ambrosetti, J. Funct. Anal. 14 pp 349– (1973)
[4] Benci, Arch. Rational Mech. Anal. 114 pp 79– (1991)
[5] and , Representation formulas for solutions to {\(\Delta\)}u – u = 0 in \(\mathbb{R}\), pp. 249–263 in: Studies in Partial Differential Equations, ed., Studies in Mathematics No. 23, Mathematical Association of America, 1982.
[6] Chen, Comm. Partial Differential Equations 16 pp 1549– (1991)
[7] Esteban, Proc. Roy. Soc. Edinburgh Sect. A 93 pp 1– (1982) · Zbl 0506.35035 · doi:10.1017/S0308210500031607
[8] Friedman, Arch. Rational Mech. Anal. 52 pp 134– (1973)
[9] , and , Symmetry of positive solutions of nonlinear elliptic equations in \(\mathbb{R}\), pp. 369–402 in: Mathematical Analysis and Applications, Part A, ed., Adv. Math. Suppl. Studies 7A, Academic Press, New York, 1981.
[10] and , Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin, 1983. · Zbl 0361.35003 · doi:10.1007/978-3-642-61798-0
[11] Kwong, Differential Integral Equations 4 pp 583– (1991)
[12] Lin, J. Differential Equations 72 pp 1– (1988)
[13] Generalized Solutions of Hamilton-Jacobi Equations, Pitman Research Notes Series No. 69, Pitman, London, 1982.
[14] Ni, Comm. Pure Appl. Math. 44 pp 819– (1991)
[15] Ni, Duke Math. J. 70 pp 247– (1993)
[16] Wang, Arch. Rational Mech. Anal. 120 pp 375– (1992)
[17] Locating the blow-up point of a semilinear Dirichlet problem involving critical Sobolev exponent, preprint.
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