Multi-bump solutions for a class of quasilinear problems involving variable exponents

We establish the existence of multi-bump solutions for the following class of quasilinear problems $$- \Delta_{ p(x) } u + \big( \lambda V(x) + Z(x) \big) u ^{ p(x)-1 } = f(x,u) \text{ in } \mathbb R^N, \, u \ge 0 \text{ in } \mathbb R^N,$$ where the nonlinearity $f \colon \mathbb R^N \times \mathbb R \to \mathbb R$ is a continuous function having a subcritical growth and potentials $V, Z \colon \mathbb R^N \to \mathbb R$ are continuous functions verifying some hypotheses. The main tool used is the variational method.