Stationary quantum Zakharov systems involving a higher competing perturbation

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.

Keywords

• quantum zakharov system
• variational methods
• multiple solutions