Year: 2010
Journal of Partial Differential Equations, Vol. 23 (2010), Iss. 3 : pp. 222–234
Abstract
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 > 0, where ω(τ^2)τ→+∞ as τ → 0 and,λ > 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jpde.v23.n3.2
Journal of Partial Differential Equations, Vol. 23 (2010), Iss. 3 : pp. 222–234
Published online: 2010-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 13
Keywords: Schrödinger equations