Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 06, pp. 1-18.

Title: Stationary quantum Zakharov systems involving a higher competing perturbation

Authors: Shuai Yao  (Shandong Univ. Technology, Zibo, China)
         Juntao Sun (Shandong Univ. Technology, Zibo, China)
         Tsung-Fang Wu (National Univ. of Kaohsiung, Kaohsiung, Taiwan)

Abstract:
 We consider the stationary quantum Zakharov system with a higher 
 competing  perturbation
 $$\displaylines{
 \Delta ^2u-\Delta u+\lambda V(x)u=K(x)u\phi
 -\mu | u|^{p-2}u \quad \text{in }\mathbb{R}^3, \cr
 -\Delta \phi +\phi =K(x)u^2 \quad \text{in }\mathbb{R}^3,
 }$$
 where $\lambda >0$, $\mu>0$, $p>4$ and functions $V$ and $K$ are both
 nonnegative. Such problem can not be studied via the common arguments in
 variational methods, since Palais-Smale sequences may not be bounded.
 Using a constraint approach proposed by us recently, we prove the existence,
 multiplicity and concentration of nontrivial solutions for the above problem.
 
Submitted July 21, 2019. Published January 10, 2020.
Math Subject Classifications: 35J35, 35B38.
Key Words: Quantum Zakharov system; variational methods; multiple solutions.