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On the shape of least-energy solutions to a semilinear Neumann problem. (English) Zbl 0754.35042
The object of study in this paper is the semilinear Neumann problem \[ d\Delta u-u+u^ p=0\;\text{ in }\Omega, \qquad \partial u/\partial\nu=0\;\text{ on }\partial\Omega \tag{1} \] where \(d\) is a (small) positive constant. The usual mountain pass lemma yields a critical value \(c_ d\) of a suitable energy functional and a corresponding critical point \(u_ d\) which is a positive solution of the above equation. The authors study the behaviour of \(u_ d\) as \(d\to 0\). The significance of \(d\) is clear if one rescales: \(v(y)=u(x_ 0+dy)\), \(x_ 0\) a maximum point of \(u\). Then \(v\) satisfies (1) with \(d=1\) but on the larger domain \({1\over d}(\Omega-x_ 0)\) so that as \(d\) approaches 0 one expects that the rescaled solutions of (1) approach the solutions of the asymptotic problem (2) \(\Delta u-u+u^ p=0\) in \(\mathbb{R}^ N\). The first result (Theorem 2.1) locates the maximum points of \(u_ d\) for \(d\) small. There is only one such point and it lies on the boundary. Equation (2) has a unique positive solution \(w\) and the second important result of this paper states that the rescalings of \(u_ d\) as above converge in some precise sense to \(w\).

35J65 Nonlinear boundary value problems for linear elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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[1] Ambrosetti, J. Funct. Anal. 14 pp 349– (1973)
[2] Berestycki, C. R. Acad. Sc. Paris, Série I Math. 297 pp 307– (1983)
[3] Berestycki, Arch. Rational Mech. Anal. 82 pp 313– (1983)
[4] and , Uniqueness of the ground state solution of {\(\Delta\)}u + f(u) = 0 in \(\mathbb{R}\)n, n 3, Comm. PDE, to appear.
[5] Coleman, Comm. Math. Phys. 58 pp 211– (1978)
[6] Ding, Arch. Rational Mech. Anal. 91 pp 283– (1986)
[7] Asymptotic results for finite energy solutions of semilinear elliptic equations, J. Diff. Eqns., to appear. · Zbl 0778.35009
[8] Fife, Arch. Rational Mech. Anal. 52 pp 205– (1973)
[9] Gierer, Kybernetik (Berlin) 12 pp 30– (1972)
[10] Gidas, Advances in Math., Supplementary Studies 7A pp 369– (1981)
[11] and , Elliptic Partial Differential Equations of Second Order, Second Edition, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983. · Zbl 0361.35003
[12] Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math. 1150, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1985. · Zbl 0593.35002
[13] Keller, J. Theor. Biol. 26 pp 399– (1970)
[14] Kwong, Arch. Rational Mech. Anal. 105 pp 243– (1989)
[15] Kwong, Differential and Integral Equations 4 pp 583– (1991)
[16] and , On the diffusion coefficient of a semilinear Neumann problem, in Calculus of Variations and Partial Differential Equations, , and , eds., Lecture Notes in Math. 1340, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1988, pp. 160–174.
[17] Lin, J. Differential Equations 72 pp 1– (1988)
[18] McLeod, Arch. Rational Mech. Anal. 99 pp 115– (1987)
[19] Uniqueness of positive radial solutions of {\(\Delta\)}u + f(u) = 0 in \(\mathbb{R}\)n, II, prepeprint.
[20] Recent progress in semilinear elliptic equations, RIMS Kokyuroku 679, Kyoto University, 1989, pp. 1–39.
[21] Takagi, J. Differential Equations 61 pp 208– (1986)
[22] Zhang, Acta Math. Sci. (English Edition) 8 pp 449– (1988)
[23] Zhu, Nonlinear Analysis 12 pp 1297– (1988)
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