### Existence of periodic solutions for some singular elliptic equations with strong resonant data

#### Abstract

We prove the existence of at least one $T$-periodic solution $(T> 0)$ for differential equations of the form

$$

\left(\frac{u'(t)}{\sqrt{1-{u'}^2(t)}}\right)' =f(u(t))+h(t),\quad \text{in } (0,T),

$$

where

$f$ is a continuous function defined on $\mathbb{R}$ that satisfies a {\it strong resonance condition}, $h$ is continuous and with zero mean value. Our method uses variational techniques for nonsmooth functionals.

$$

\left(\frac{u'(t)}{\sqrt{1-{u'}^2(t)}}\right)' =f(u(t))+h(t),\quad \text{in } (0,T),

$$

where

$f$ is a continuous function defined on $\mathbb{R}$ that satisfies a {\it strong resonance condition}, $h$ is continuous and with zero mean value. Our method uses variational techniques for nonsmooth functionals.

#### Keywords

$\Phi$-laplacian; strong resonance condition; periodic solutions

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