Positive, negative, and sign-changing solutions to a quasilinear Schrödinger equation with a parameter

Abstract

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.

Infinitely Many Sign-Changing Solutions for the Nonlinear Schrödinger-Poisson System with Super 2-linear Growth at Infinity

“…Many works were presented on studying the existence, multiplicity of the solutions to Eq. (1.5), see [24,33,35,43,44] and the references therein. Compared to the Schrödinger equation, i.e., λ = 0, the case λ > 0 with a nonlocal term λφ(x)u becomes more difficult.…”

Infinitely many solutions to a quasilinear Schrödinger equation with a local sublinear term

“…Moreover, the proof of (2.8) implies that {λ n } is bounded from above. By claim 2 of Lemma 2.2 in [45], up to a subsequence, we may assume that λ n → λ 0 , and…”

Least energy sign-changing solutions for Schrödinger-Poisson system with critical growth

“…Many works presented on studying the existence, multiplicity of the solutions to equation (1.3), see [12,14,16,25,26] and the references therein. Noting that system (1.2) involves a nonlocal term λφ(x)u, which brings some properties different from the case λ ≡ 0.…”

“…In recent years, more and more researchers are paying attention to Schrödinger-Poisson system, especially on the existence of positive solutions, multiple solutions, least energy solutions, such as [2,7,10,17,21,26,27] and the references therein. To our knowledge, about the existence of sign-changing solutions for subcritical Schrödinger-Poisson system, there are few works presented in [8,9,19,23].…”