Multiplicity of solutions for a p-Schrödinger–Kirchhoff-type integro-differential equation

Abstract

We consider the integro-differential problem (P):

$$\begin{aligned} -\left( a+b\left( \int _{\mathbb {R}^{N}}|\nabla u|^{p} \textrm{d} x\right) ^{p-1}\right) \Delta _{p} u +V(x)|u|^{p-2} u =f(x,u), \quad x\in \mathbb {R}^{N}, \end{aligned}$$

with $|u(x)| \longrightarrow 0$, as $|x| \longrightarrow +\infty$. We assume that $a, b>0$, $N \ge 2$, $1<p< N < +\infty$, $V\in \textrm{C}^{}(\mathbb {R}^{N})$ with $\inf (V)>0$, and that $f:\mathbb {R}^{N}\times \mathbb {R}\longrightarrow \mathbb {R}$ verifies conditions introduced by Duan and Huang. We prove the existence of a non-trivial ground state solution and, by a Ljusternik–Schnirelman scheme, the existence of infinitely many non-trivial solutions.