Existence of nontrivial solutions for Schrödinger-Kirchhoff type equations involving the fractional <i>p</i>-Laplacian and local nonlinearity

Abstract

In this paper, we deal with the existence of nontrivial solutions for the following Kirchhoff-type equation $ M\left(\,\,\displaystyle{\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\text{d}x\text{d}y}\right)(-\Delta)_{p}^{s} u+V(x)|u|^{p-2}u = \lambda f(x,u),\,\, \text{in}\,\,\mathbb{R}^N, $ where $0 0$ is a real parameter, $(-\Delta)_{p}^{s}$ is the fractional $p$-Laplacian operator, $V:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is a potential function, $M$ is a Kirchhoff function, the nonlinearity $f:\mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function and just super-linear in a neighborhood of $u = 0$. By using an appropriate truncation argument and the mountain pass theorem, we prove the existence of nontrivial solutions for the above equation, provided that $\lambda$ is sufficiently large. Our results extend and improve the previous ones in the literature.