Schrödinger equations in $\mathbb{R}^4$ involving the biharmonic operator with critical exponential growth

Abstract

In this paper we prove the existence of at least one nontrivial ground state solution for fourth-order elliptic equations of the form

$${\mathrm{\Delta }}^{2}u-\mathrm{\Delta }u+u=K\left(x\right)\left[f\left(u\right)+g\left(u\right)\right],x\in {ℝ}^4,u\in H^2\left({ℝ}^4\right),$$($P$)

where ${\mathrm{\Delta }}^2:=\mathrm{\Delta }\left(\mathrm{\Delta }\right)$ is the biharmonic operator, $f$ is a continuous nonnegative function with polynomial growth at infinity, $g$ is a continuous nonnegative function with exponential growth and $K$ is a positive bounded continuous function that can vanish at infinity. Our results complete the analysis made in F. Sani (Comm. Pure Appl. Anal. 12 (2013), 405–428), where the author studied Schrödinger equations involving the biharmonic operator with coercive potentials. Our approach is based on various techniques such as the mountain-pass theorem, Trundinger–Moser type inequalities and compactness results.