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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.

##### MSC:
 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
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##### References:
 [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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