Abstract

We are interested in a general Choquard equation −Δ+m2u−mu+V(x)u−μ|x|u=∫RNF(y,u(y))|x−y|N−αdyf(x,u)−K(x)|u|q−2u under suitable assumptions on the bounded potential V and on the nonlinearity f. Our analysis extends recent results by the second and third author on the problem with μ=0 and pure-power nonlinearity f(x,u)=|u|p−2u. We show that, under appropriate assumptions on the potential, whether the ground state does exist or not. Finally, we study the asymptotic behaviour of ground states as μ→0+.

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