Nontrivial solutions and least energy nodal solutions for a class of fourth-order elliptic equations

This paper is concerned with the following fourth-order elliptic equation

$$begin{aligned} left{ begin{array}{ll} displaystyle Delta ^{2}u-Delta u+V(x)u=|u|^{p-1}u,,mathrm{in},mathbb {R}^{N}, uin H^{2}left( mathbb {R}^{N}right) , end{array} right. end{aligned}$$

where (pin (2,,2_{*}-1),,u{text {:}},mathbb {R}^{N}rightarrow mathbb R.) Under some appropriate assumptions on potential V(x),  the existence of nontrivial solutions and the least energy nodal solution are obtained by using variational methods.