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Abstract

We consider Kirchhoff type problems of the form $$ \left\{\begin{aligned} &{-} M(\rho(u)) (\mathrm{div}(a(|\nabla u|)\nabla u)-a(|u|)u)=K(x)f(u) \quad \text{in } \Omega,\\&\textstyle \frac{\partial u}{\partial \nu} = 0 \quad \text{on } \partial\Omega, \end{aligned}\right. $$ where $\Omega \subset \mathbb R^N$, $N \geq 3$, is a smooth bounded domain, $\nu$ is the outward unit normal to $\partial\Omega$, $\rho(u)= \int_\Omega (\Phi (|\nabla u|)+\Phi(|u|) )\, dx$, $M: [0,\infty) \to \mathbb R$ is a continuous function, $K\in L^\infty(\Omega)$, and $f: \mathbb R\to\mathbb R$ is a continuous function not satisfying the Ambrosetti–Rabinowitz type condition. Using variational methods, we obtain some existence and multiplicity results.