Multiple normalized solutions for Choquard equation involving the biharmonic operator and competing potentials in ${ℝ}^N$

This paper is concerned with the existence of multiple normalized solutions for a class of Choquard equation involving the biharmonic operator and competing potentials in ${ℝ}^N$:

where ${\mathrm{\Delta }}^2$ is the biharmonic operator, $c,𝜀>0$, $\mu \in (0,N)$, $\lambda \in ℝ$ is an unknown parameter that appears as the Lagrange multiplier, the absorption potential $V$, reaction potential $G$ and nonlinear term $f$ are continuous functions and satisfy some assumptions. With the help of the minimization techniques and Lusternik–Schnirelmann category, we show that the number of normalized solutions is not less than the number of global minimum points of $V$ when the parameter $𝜀$ is sufficiently small. Furthermore, and more interestingly, we can prove the existence of multiple solutions for this problem by using the Morse theory. As far as we know, this study seems to be the first contribution regarding the concentration behavior for Choquard equation involving the biharmonic operator.