TY - JOUR
T1 - Positive Solutions for Singular Quasilinear Schrodinger Equations with One Parameter, II
JO - Journal of Partial Differential Equations
VL - 3
SP - 222
EP - 234
PY - 2010
DA - 2010/08
SN - 23
DO - http://doi.org/10.4208/jpde.v23.n3.2
UR - https://global-sci.org/intro/article_detail/jpde/5231.html
KW - Schrödinger equations
KW - solitary waves
KW - variational methods
AB - <p>We establish the existence of positive bound state solutions for the singular quasilinear Schrödinger equation i\frac{∂ψ}{∂t}=-div(ρ(|∇ψ|^2)∇ψ)+ω(|ψ|^2)ψ-λρ(|ψ|^2)ψ, x∈Ω, t &gt; 0, where ω(τ^2)τ→+∞ as τ → 0 and,λ &gt; 0 is a parameter and Ω is a ball in R^N. This problem is studied in connection with the following quasilinear eigenvalue problem with Dirichlet boundary condition -div(ρ(|∇Ψ|^2)∇Ψ)=λ_1ρ(|Ψ|^2)Ψ, x∈Ω. Indeed, we establish the existence of solutions for the above Schrödinger equation when λ belongs to a certain neighborhood of the first eigenvalue λ_1 of this eigenvalue problem. Themain feature of this paper is that the nonlinearity ω(|ψ|^2)ψ is unbounded around the origin and also the presence of the second order nonlinear term. Our analysis shows the importance of the role played by the parameter λ combined with the nonlinear nonhomogeneous term div(ρ(|∇ψ|^2)∇ψ) which leads us to treat this problem in an appropriateOrlicz space. The proofs are based on various techniques related to variational methods and implicit function theorem.</p>