No abstract

In this paper, we consider the existence and multiplicity of solutions for the following quasilinear Choquard equation: −Δu+V(x)u−uΔ(u2)=(|x|−μ*|u|p)|u|p−2u,   x∈RN, where N ≥ 3, μ∈(0,N+22), p∈(2,4N−4μN−2). Under some assumptions on V, we obtain the existence of positive solutions, negative solutions, and high-energy solutions via perturbation method.

In this paper, we study the following quasilinear Schrödinger equation with a parameter: −Δu+V(x)u−καΔ(|u|2α)|u|2α−2u=|u|p−2u+|u|(2α)2*−2u  in  RN, where N ≥ 3, α>12, 2 < p < (2α)2*, and κ is a positive constant. Under different assumptions on V, we obtain the existence of positive, negative, and sign-changing solutions. Our results generalize the results of Liu et al. [J. Differ. Equations 187, 473–493 (2003)] into the critical case for general α.