An asymptotic estimate of the number of bifurcating solutions for the equation $-\Delta u=\mu f(u)$

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 $$\liminf_{\varepsilon \rightarrow 0} \ \alpha_{\varepsilon}(L) \geq \frac{5}{8} \pi L + O(\sqrt{L}) \quad as \ L \rightarrow \infty.$$