Existence of a nonnegative bound state for a higher-order logarithmic Schrödinger equation

In this paper, we study the existence of nonnegative bound states for the higher-order logarithmic Schrödinger equation $$ -αΔu+V(x)u-βu\ln|u| = γu(\ln|u|)^m, \qquad u\in H^1(\mathbb R^N), $$ where $α,β,γ>0$, $m$ is an odd positive integer and $V$ is a positive bounded potential converging to a cons