Stationary quantum Zakharov systems involving a higher competing perturbation

Abstract

We consider the stationary quantum Zakharov system with a higher competing perturbation Δ2u - Δu + λV(x)u = K(x)uφ - μ|u|p-2u in ℝ3, -Δφ + φ = K(x)u2 in ℝ3, where λ > 0, μ > 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.