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 $- Δ_{ p(x) } u + \big( λV(x) + Z(x) \big) u ^{ p(x)-1 } = f(x,u) \text{ in } \mathbb R^N, \, u \ge 0 \text{ in } \mathbb R^N,$ 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.

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