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2000 An asymptotic estimate of the number of bifurcating solutions for the equation $-\Delta u=\mu f(u)$
W. Krawcewicz, W. Marzantowicz
Funct. Approx. Comment. Math. 28(none): 195-200 (2000). DOI: 10.7169/facm/1538186695

Abstract

In this paper we present a lower estimate on the number of non-zero solutions $(u,\mu)$ of the following boundary value problem $$\left\{ \begin{array}{l@{\quad \quad \quad}l} -\Delta u=\mu \cdot f(u) & \mathrm{on} & \Omega \\ u \equiv 0 & \mathrm{on} & \partial\Omega \end{array} \right. \qquad \qquad (\mathcal{P})$$ where $\mu \in \mathbb{R}, \Omega = (-\pi /2;\pi /2)^2$ and $f : \mathbb{R} \rightarrow \mathbb{R}$ is a function of class $C^1$ satisfying some additional requirements. By using the symmetry properties of the problem $(\mathcal{P})$ and classical results from number theory, we show that the numbers $\alpha_{\varepsilon}(L)$ of all distinct nontrivial solutions $(u, \mu)$ of $(\mathcal{P})$ such that $\|u\| \lt \varepsilon$, for $\varepsilon > 0$, where $0 \lt \mu \lt L + 1$, satisfy the following inequality $$\liminf_{\varepsilon \rightarrow 0} \ \alpha_{\varepsilon}(L) \geq \frac{5}{8} \pi L + O(\sqrt{L}) \quad as \ L \rightarrow \infty.$$

Dedication

Dedicated to Włodzimierz Staś on the occasion of his 75th birthday

Citation

Download Citation

W. Krawcewicz. W. Marzantowicz. "An asymptotic estimate of the number of bifurcating solutions for the equation $-\Delta u=\mu f(u)$." Funct. Approx. Comment. Math. 28 195 - 200, 2000. https://doi.org/10.7169/facm/1538186695

Information

Published: 2000
First available in Project Euclid: 29 September 2018

zbMATH: 0977.35051
MathSciNet: MR1824004
Digital Object Identifier: 10.7169/facm/1538186695

Subjects:
Primary: 58E09
Secondary: 11Lxx, 35J50, 35J60, 35J65

Rights: Copyright © 2000 Adam Mickiewicz University

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