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Volume 19, Issue 3
Semiclassical States of Hamiltonian System of Schrodinger Equations with Subcritical and Critical Nonlinearities

Yanheng Ding & Fanghua Lin

J. Part. Diff. Eq., 19 (2006), pp. 232-255.

Published online: 2006-08

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  • Abstract

We consider the system of perturbed Schrödinger equations {-ε²Δφ + α(x)φ = β(x)ψ + F_ψ(x, φ, ψ) -ε²Δψ + α(x)ψ = β(x)φ + F_φ(x, φ, ψ) w := (φ, ψ) ∈ H¹(\mathbb{R}^N, \mathbb{R}²) where N ≥ 1, α and β are continuous real functions on \mathbb{R}^N, and F : \mathbb{R}^N × \mathbb{R}² → \mathbb{R} is of class C¹. We assume that either F(x, w) is super-quadratic and subcritical in w ∈ \mathbb{R}² or it is of the form ∼ \frac{1}{p}P(x)|w|^p + \frac{1}{2∗}K(x)|w|2∗ with p ∈ (2, 2∗) and 2∗ = 2N/(N - 2), the Sobolev critical exponent, P(x) and K(x) are positive bounded functions. Under proper conditions we show that the system has at least one nontrivial solution w_ε provided ε ≤ ξ and for any m ∈ \mathbb{N}, there are m pairs of solutions w_ε provided that ε ≤ ξ_m and that F(x, w) is, in addition, even in w. Here ξ and ξ_m are sufficiently small positive numbers. Moreover, the energy of w_ε tends to 0 as ε → 0.

  • Keywords

Perturbed Schrödinger equation critical nonlinearity multiple solutions

  • AMS Subject Headings

58E05 58E50.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-19-232, author = {}, title = {Semiclassical States of Hamiltonian System of Schrodinger Equations with Subcritical and Critical Nonlinearities}, journal = {Journal of Partial Differential Equations}, year = {2006}, volume = {19}, number = {3}, pages = {232--255}, abstract = {

We consider the system of perturbed Schrödinger equations {-ε²Δφ + α(x)φ = β(x)ψ + F_ψ(x, φ, ψ) -ε²Δψ + α(x)ψ = β(x)φ + F_φ(x, φ, ψ) w := (φ, ψ) ∈ H¹(\mathbb{R}^N, \mathbb{R}²) where N ≥ 1, α and β are continuous real functions on \mathbb{R}^N, and F : \mathbb{R}^N × \mathbb{R}² → \mathbb{R} is of class C¹. We assume that either F(x, w) is super-quadratic and subcritical in w ∈ \mathbb{R}² or it is of the form ∼ \frac{1}{p}P(x)|w|^p + \frac{1}{2∗}K(x)|w|2∗ with p ∈ (2, 2∗) and 2∗ = 2N/(N - 2), the Sobolev critical exponent, P(x) and K(x) are positive bounded functions. Under proper conditions we show that the system has at least one nontrivial solution w_ε provided ε ≤ ξ and for any m ∈ \mathbb{N}, there are m pairs of solutions w_ε provided that ε ≤ ξ_m and that F(x, w) is, in addition, even in w. Here ξ and ξ_m are sufficiently small positive numbers. Moreover, the energy of w_ε tends to 0 as ε → 0.

}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5330.html} }
TY - JOUR T1 - Semiclassical States of Hamiltonian System of Schrodinger Equations with Subcritical and Critical Nonlinearities JO - Journal of Partial Differential Equations VL - 3 SP - 232 EP - 255 PY - 2006 DA - 2006/08 SN - 19 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5330.html KW - Perturbed Schrödinger equation KW - critical nonlinearity KW - multiple solutions AB -

We consider the system of perturbed Schrödinger equations {-ε²Δφ + α(x)φ = β(x)ψ + F_ψ(x, φ, ψ) -ε²Δψ + α(x)ψ = β(x)φ + F_φ(x, φ, ψ) w := (φ, ψ) ∈ H¹(\mathbb{R}^N, \mathbb{R}²) where N ≥ 1, α and β are continuous real functions on \mathbb{R}^N, and F : \mathbb{R}^N × \mathbb{R}² → \mathbb{R} is of class C¹. We assume that either F(x, w) is super-quadratic and subcritical in w ∈ \mathbb{R}² or it is of the form ∼ \frac{1}{p}P(x)|w|^p + \frac{1}{2∗}K(x)|w|2∗ with p ∈ (2, 2∗) and 2∗ = 2N/(N - 2), the Sobolev critical exponent, P(x) and K(x) are positive bounded functions. Under proper conditions we show that the system has at least one nontrivial solution w_ε provided ε ≤ ξ and for any m ∈ \mathbb{N}, there are m pairs of solutions w_ε provided that ε ≤ ξ_m and that F(x, w) is, in addition, even in w. Here ξ and ξ_m are sufficiently small positive numbers. Moreover, the energy of w_ε tends to 0 as ε → 0.

Yanheng Ding & Fanghua Lin . (2019). Semiclassical States of Hamiltonian System of Schrodinger Equations with Subcritical and Critical Nonlinearities. Journal of Partial Differential Equations. 19 (3). 232-255. doi:
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