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.

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.

Reviewer: D.Bobrowski (Poznań)

##### 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 |

##### Keywords:

domain shape; Dirichlet problem; number of positive solutions; Neumann boundary value problem
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\textit{E. N. Dancer}, J. Differ. Equations 87, No. 2, 316--339 (1990; Zbl 0729.35050)

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