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Superlinear elliptic Dirichlet problems in almost spherically symmetric exterior domains. (English) Zbl 0664.35028

The authors study the existence of positive solutions of the problem \[ \Delta u-u+f(u)=0 \text{ in } \Omega;\quad u=0 \text{ on } \partial \Omega, \tag{1} \] where \(\Omega\) is an exterior domain in \(R^3\) (the complement of a nearly spherical domain) and f is a “superlinear” function vanishing at zero to order higher than the first. This problem gives rise to the consideration of two additional questions: the structure of positive, radially symmetric solutions of (1) which vanish at infinity and the study of the properties of the spectrum of the eigenvalue problem \[ \Delta v-v-\lambda f'(u)v=0 \text{ in } \Omega_ R, \quad v\in \overset\circ W_{1,2}(\Omega_ R); \quad v=0 \text{ on } \partial \Omega_ R, \] where \(\Omega_ R=\{x\in R^ 3: | x| >R\}\) and \(u\) is a positive radially symmetric solution of (1) with \(\Omega =\Omega_ R.\)
For the proof of their results, the authors use a perturbation argument and the approach to the eigenvalue problem is realized using the expansion of v into a series of spherical harmonics.
Reviewer: A.Cañada

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35P05 General topics in linear spectral theory for PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35A30 Geometric theory, characteristics, transformations in context of PDEs
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[1] Berestycki, H., &. P.-L. Lions, ”Nonlinear Scalar Field Equations, I. Existence of a ground state, II. Existence of infinitely many solutions,” Ar · Zbl 0556.35046 · doi:10.1007/BF00250555
[2] Coffman, C. V., & M. Marcus, ”Existence theorems for superlinear elliptic Dirichlet problems in exterior domains,” Nonlinear Functional Analysis, and Its Applications, Proceedings of Symposia in Pure Mathematics, Providence 1986, Vol. 45, Amer. Math. Soc. · Zbl 0596.35048
[3] Esteban, M. J., & P.-L. Lions, ”Existence and nonexistence results for semilinear elliptic problems in unbounded domains,” Pr · Zbl 0506.35035 · doi:10.1017/S0308210500031607
[4] Hartman, P., Ordinary Differential Equations, S. Hartman, Baltimore, 1973. · Zbl 0281.34001
[5] Kato, T., Perturbation Theory for Linear Operators, Springer-Verlag, New York 1976. · Zbl 0342.47009
[6] Lions, P.-L., ”Symétrie et compacité dans les espaces de Sobolev · Zbl 0501.46032 · doi:10.1016/0022-1236(82)90072-6
[7] Müller, C., Spherical Harmonies, Springer-Verlag, Lecture Notes in Mathematics 17, New York 1966.
[8] Nehari, Z., ”On a class of second order differential equations,” · Zbl 0097.29501 · doi:10.1090/S0002-9947-1960-0111898-8
[9] Nehari, Z., ”On a nonlinear differential equation arising in nuclear physics,” · Zbl 0124.30204
[10] Noussair, E. S., & C. A. Swanson, ”Positive solutions of semilinear elliptic problems in unbounded domain · Zbl 0583.35039 · doi:10.1016/0022-0396(85)90061-0
[11] Strauss, W. A., ”Existence of solitary waves in higher dimensio · Zbl 0356.35028 · doi:10.1007/BF01626517
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