Ting Liu
In this paper, we consider the following higher-order Schrödinger equation with critical growth: \[ (-\Delta )^mu+V(y)u=Q(y)u^{m^*-1}, \quad u>0 \ \hbox{in} \ \mathbb{R}^{N}, \ u \in \mathcal{D}^{m,2}\,(\mathbb{R}^{N}), \] where $m^*=\frac{2N}{N-2m},\; N\geq 4m+1$, $m \geq 2$ is an integer, $(y',y'') \in \mathbb{R}^{2} \times \mathbb{R}^{N-2}$ and $V(y) = V(|y'|,y'')$ and $Q(y) = Q(|y'|,y'')$ are bounded non-negative functions in $\mathbb{R}^{+} \times \mathbb{R}^{N-2}$. By using finite dimensional reduction argument and local Pohozaev type identities, we show that if $N \geq 4m+1$, $Q(r,y'')$ has a critical point $(r_0,y_0'')$ satisfying $r_0 >0$, $Q(r_0,y_0'') > 0$, $ D^{\alpha}Q(r_0,y_0'')=0,$ $|\alpha| \leq 2m-1$ and $ {\rm deg} (\nabla (Q(r,y'')), (r_0,y_0'')) \neq 0$, and $\frac{1}{(2m-1)!m^*} \sum_{|\alpha|=2m}D^{\alpha}Q(r_0,y_0'') \int_{\mathbb{R}^N}y^{\alpha}U_{0,1}^{m^*} -m V(r_0,y_0'') \int_{\mathbb{R}^{N}} U_{0,1}^2 < 0$, then the above problem has infinitely many solutions, whose energy can be arbitrarily large. Different from that in [Commun. Contemp. Math., 2022, 24: Paper No. 2050071], in our case, the higher order derivative of $Q(r_0,y_0'')$ plays an important role in the construction of bubble solutions. Besides, it implies that the potential $V(r_0,y_0'') $ can influence the sign of higher order derivative of $Q(r_0,y_0'')$.
