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Volume 23, Issue 3
Positive Solutions for Singular Quasilinear Schrodinger Equations with One Parameter, II

Abbas Moameni & Daniel C. Offin

DOI: 10.4208/jpde.v23.n3.2

J. Part. Diff. Eq., 23 (2010), pp. 222-234.

Published online: 2010-08

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  • 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.

  • Keywords

Schrödinger equations solitary waves variational methods

  • AMS Subject Headings

35J10 35J20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-23-222, author = {Abbas Moameni and Daniel C. Offin }, title = {Positive Solutions for Singular Quasilinear Schrodinger Equations with One Parameter, II}, journal = {Journal of Partial Differential Equations}, year = {2010}, volume = {23}, number = {3}, pages = {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.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v23.n3.2}, url = {http://global-sci.org/intro/article_detail/jpde/5231.html} }
TY - JOUR T1 - Positive Solutions for Singular Quasilinear Schrodinger Equations with One Parameter, II AU - Abbas Moameni & Daniel C. Offin 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 -

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.

Abbas Moameni and Daniel C. Offin . (2010). Positive Solutions for Singular Quasilinear Schrodinger Equations with One Parameter, II. Journal of Partial Differential Equations. 23 (3). 222-234. doi:10.4208/jpde.v23.n3.2
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