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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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