## Differential and Integral Equations

### Existence and multiplicity of solutions of some conservative pendulum-type equations with homogeneous Dirichlet conditions

#### Abstract

Let us consider the resonant boundary value problem \begin{align} - &u''(x) - u(x) + g(u(x)) = h(x), \quad x \in [0,\pi], \\ &u(0) = u(\pi) = 0, \end{align} where $g: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous and $T$-periodic function with zero mean value, not identically zero, and $h \in C[0,\pi].$ If each $h \in C[0,\pi]$ is written as $h(x) = a\sin x + \tilde{h}(x),$ where $a \in \mathbb{R}$ and $\int_{0}^{\pi} \tilde{h}(x) \sin x \ dx = 0,$ then, it is shown that for each $\tilde{h},$ there are real numbers $a_{1}(\tilde{h}) < 0 < a_{2}(\tilde{h})$ (which depend continuously on $\tilde{h}$), such that there is solution if and only if $a\in [a_{1}(\tilde{h}),a_{2}(\tilde{h})].$ In relation to the multiplicity, it is proved that the number of solutions increases to infinity as $a$ goes to zero. The proof combines different tools such as Liapunov-Schmidt reduction and upper-lower solutions notions, together with a careful analysis of the connected subsets of the solution set of the auxiliary equation in the alternative method, as well as a detailed study of the oscillatory behavior of some integrals associated to the bifurcation equation of the previous problem.

#### Article information

Source
Differential Integral Equations, Volume 10, Number 6 (1997), 1113-1122.

Dates
First available in Project Euclid: 1 May 2013

https://projecteuclid.org/euclid.die/1367438222

Mathematical Reviews number (MathSciNet)
MR1608041

Zentralblatt MATH identifier
0938.34013

#### Citation

Cañada, A.; Roca, F. Existence and multiplicity of solutions of some conservative pendulum-type equations with homogeneous Dirichlet conditions. Differential Integral Equations 10 (1997), no. 6, 1113--1122. https://projecteuclid.org/euclid.die/1367438222