A non-trivial solution for a p-Schrödinger–Kirchhoff-type integro-differential system by non-smooth techniques

We consider the integro-differential system \((\textrm{P}_m)\):

$$\begin{aligned} - \left( a_k+b_k \left( \displaystyle \int _{{\mathbb {R}}^{N}} |\nabla u_k|^{p} dx \right) ^{p-1} \right) \Delta _{p} u_k + V(x) |u_k|^{p-2} u_k = \partial _{k} F(u_1,\ldots ,u_m), \end{aligned}$$

where \(x\in {\mathbb {R}}^N\), \(a_k>0\), \(b_k\ge 0\), \(N\ge 2\) and \(1<p<N\), \(u_k \in \textrm{W}^{1,p}({\mathbb {R}}^{N})\), for \(k=1,\ldots ,m\). By \(\partial _{k} F(u_1,\ldots ,u_m),\) it is denoted the k-th partial generalized gradient in the sense of Clarke. The potential \(V\in \textrm{C} \left( {\mathbb {R}}^N \right) \) verifies \(\inf (V)>0\) and a coercivity property introduced by Bartsch et al. The coupling function \(F:{\mathbb {R}}^m\longrightarrow {\mathbb {R}}\) is locally Lipschitz and verifies conditions introduced by Duan and Huang. By applying tools from the non-smooth critical point theory, we prove the existence of a non-trivial mountain pass solution of \((\textrm{P}_m)\).