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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 = {}, 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 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 & Daniel C. Offin . (2019). 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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