Sign-changing solutions for a parameter-dependent quasilinear equation

Abstract

We consider quasilinear elliptic equations, including the following Modified Nonlinear Schrödinger Equation as a special example:    ∆u + 1 2 u∆u 2 + λ|u| r−2 u = 0, in Ω, u = 0 on ∂Ω, where Ω ⊂ R N (N ≥ 3) is a bounded domain with smooth boundary, λ > 0, r ∈ (2, 4). We prove as λ becomes large the existence of more and more sign-changing solutions of both positive and negative energies.

Nodal solutions for quasilinear Schrödinger equations with asymptotically 3-linear nonlinearity

“…[23] obtained a least energy sign-changing solution and the proof was based on the method of Nehari manifold, deformation arguments and L ∞ -estimates. Later, Liu et al [15] considered a parameter-dependent quasilinear equation ∆u + 1 2 u∆(u 2 ) + λ|u| r−2 u = 0, in Ω, u = 0, on ∂Ω, where r ∈ (2,4). By means of a perturbation approach, they proved the existence of more and more sign-changing solutions with both positive and negative energies when λ becomes large.…”

“…Observe that no information about nodal domains is available for sign-changing solutions obtained in [15,24] for quasilinear Schrödinger equation with sub-cubic or cubic nonlinearity. In this paper, we are interested in sign-changing solutions with nodal domains.…”

Research Progress in the Pharmacological Effects and Synthesis of Nicotine

“…Liu and co-workers reported the methyl nicotinate ( 12) was reacted with di-tert-butyl succinate (85), and then acidified to obtain 4-oxo-4-(pyridin-3-yl)butanoic acid (8). [98] (S)-5-(pyridin-3-yl)dihydrofuran-2(3H)-one ( 86) was given via the asymmetric reduction by R-(+)-2-methyl-CBS-oxazaborolidine. Then 86 was reacted with methylamine hydrobromide to obtain (S)-9.…”