Positive Solutions for Singular Quasilinear Schrodinger Equations with One Parameter, II (Journal of Partial Differential Equations, 2010)
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.