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The effect of domain shape on the number of positive solutions of certain nonlinear equations. II. (English) Zbl 0729.35050
[For part I see ibid. 74, No.1, 120-156 (1988; Zbl 0662.34025).]
This paper is a continuation of part I. The author deals with the problem \[ (*)\quad -\Delta u=\lambda f(u)\text{ in } \Omega \subset {\mathbb{R}}^ m;\quad u=0\text{ on } \partial \Omega. \] Changing the domain, he studies the number of positive solutions to the problem (*). If \(\Omega\) is star-shaped then the equation (*) has many solutions. In some cases the positive solutions are not connected.
Next, using the same technic as in part I the author proves some results for a non-selfadjoint problem, for the Neumann boundary value problem and for the parabolic problem.

35J65 Nonlinear boundary value problems for linear elliptic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
Full Text: DOI
[1] Adams, R, Sobolev spaces, (1975), Academic Press New York · Zbl 0314.46030
[2] Amann, H, Nonlinear eigenvalue problems having precisely two solutions, Math. Z., 150, 27-37, (1976) · Zbl 0329.35027
[3] \scH. Amann, Dynamic theory of quasilinear parabolic equations, II, reaction-diffusion systems, to appear. · Zbl 0729.35062
[4] Anselone, N, Collectively compact operator approximation theory and applications to integral equations, (1971), Prentice-Hall Englewood Cliffs, NJ · Zbl 0228.47001
[5] Cosner, C; Lazer, A, Stable coexistence in the Volterra-Lotka competing species model with diffusion, SIAM J. appl. math., 44, 1112-1134, (1984) · Zbl 0562.92012
[6] Crandall, M; Rabinowitz, P, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. rational mech. anal., 58, 207-218, (1975) · Zbl 0309.35057
[7] Dancer, E.N, The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. differential equations, 74, 120-156, (1988) · Zbl 0662.34025
[8] Dancer, E.N, A note on an equation with critical exponent, Bull. London math. soc., 20, 600-602, (1988) · Zbl 0646.35027
[9] \scE. N. Dancer, On the influence of domain shape on the existence of the large solutions of some superlinear problems, submitted for publication. · Zbl 0699.35103
[10] Dancer, E.N, On the indices of fixed points of mappings in cones and applications, J. math. anal. appl., 91, 131-151, (1983) · Zbl 0512.47045
[11] Dancer, E.N, On positive solutions of some pairs of differential equations, Trans. amer. math. soc., 284, 729, (1984) · Zbl 0524.35056
[12] Dancer, E.N; Hess, P, On stable solutions of quasilinear periodic-parabolic problems, Ann. scuola norm. sup. Pisa, 14, 123-141, (1987) · Zbl 0697.35072
[13] Dancer, E.N; Schmitt, K, On positive solutions of semilinear elliptic equations, (), 445-452 · Zbl 0661.35031
[14] Deimling, K, Nonlinear functional analysis, (1985), Springer-Verlag Berlin · Zbl 0559.47040
[15] Gelbaum, B; Olmsted, J, Counterexamples in analysis, (1964), Holden-Day San Francisco · Zbl 0121.28902
[16] Gilbarg, D; Trudinger, N, Elliptic partial differential equations of second order, (1977), Springer-Verlag Berlin · Zbl 0361.35003
[17] Hedberg, L, Spectral synthesis and stability in Sobolev spaces, (), 73-80
[18] Henry, D, Geometric theory of semilinear parabolic equations, () · Zbl 0456.35001
[19] Hess, P, On positive solutions of semilinear parabolic problems, (), 1001-1014 · Zbl 1410.41034
[20] Jimbo, S, Singular perturbation of domains and semilinear elliptic equations, J. fac. sci. univ. Tokyo sect. IA math., 35, 27-76, (1988) · Zbl 0667.35002
[21] \scS. Jimbo, Singular perturbation of domains and semilinear elliptic equation II, J. Differential Equations, to appear. · Zbl 0667.35003
[22] \scS. Jimbo, A construction of the perturbed solution of semilinear elliptic equation in the singularly perturbed domain, submitted for publication. · Zbl 0697.35013
[23] Kato, T, Perturbation theory for linear operators, (1966), Springer-Verlag Berlin · Zbl 0148.12601
[24] Ladyzhenskaya, O; Solonnikov, V; Uralceva, N, Linear and quasilinear equations of parabolic type, (1981), Amer. Math. Soc Providence
[25] Matano, H; Mimura, M, Pattern formulation in competitive-diffusion systems in non-convex domains, Publ. res. inst. math. sci., 19, 1049-1079, (1983) · Zbl 0548.35063
[26] Rauch, J; Taylor, M, Potential and scattering theory in wildly perturbed domains, J. funct. anal., 18, 27-59, (1975) · Zbl 0293.35056
[27] Saut, J; Teman, R, Generic properties on nonlinear boundary-value problems, Comm. partial differential equations, 4, 293-319, (1979) · Zbl 0462.35016
[28] Trudinger, N, Linear elliptic equations with measurable coefficients, Ann. scuola norm. sup. Pisa, 29, 265-308, (1973) · Zbl 0279.35025
[29] Vegas, J, Irregular variations of the domain in elliptic problems with Neumann boundary conditions, (), 276-287
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