Nonlinear Scalar Field Equations with L 2 Constraint: Mountain Pass and Symmetric Mountain Pass Approaches

Jun Hirata, Kazunaga Tanaka

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    2 Citations (Scopus)

    Abstract

    We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in R N {\mathbb{R}^{N}} (N ≥ 2 {N\geq 2}): { - Δ u = g (u) - μ u in R N, ≈ u ≈ L 2 (R N) 2 = m, u ∈ H 1 (R N), \displaystyle\begin{cases}-\Delta u=g(u)-\mu u\quad\text{in }\mathbb{R}^{N},% \cr\lVert u\rVert-{L^{2}(\mathbb{R}^{N})}^{2}=m,\cr u\in H^{1}(\mathbb{R}^{N})%,\end{cases} where g (ξ) ∈ C (R, R) {g(\xi)\in C(\mathbb{R},\mathbb{R})}, m > 0 {m>0} is a given constant and μ ∈ R {\mu\in\mathbb{R}} is a Lagrange multiplier. We introduce a new approach using a Lagrange formulation of problem (∗) m {(∗)-{m}}. We develop a new deformation argument under a new version of the Palais-Smale condition. For a general class of nonlinearities related to [H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I: Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 313-345], [H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 347-375], [J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in R N {\mathbb{R}^{N}}: Mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal. 35 (2010), no. 2, 253-276], it enables us to apply minimax argument for L 2 {L^{2}} constraint problems and we show the existence of infinitely many solutions as well as mountain pass characterization of a minimizing solution of the problem inf { ∫ R N 1 2 | Δ u | 2 - G (u) d x: ≈ u ≈ L 2 (R N) 2 = m }, G (ξ) = ∫ 0 ξ g (τ) τ. \inf\Bigg{\{}\int-{\mathbb{R}^{N}}{1\over 2}|{\nabla u}|^{2}-G(u)\,dx:\lVert u% \rVert-{L^{2}(\mathbb{R}^{N})}^{2}=m\Bigg{\}},\quad G(\xi)=\int-{0}^{\xi}g(% \tau)\,d\tau.

    Original languageEnglish
    JournalAdvanced Nonlinear Studies
    DOIs
    Publication statusAccepted/In press - 2019 Jan 1

    Fingerprint

    Mountain Pass
    mountains
    Scalar Field
    scalars
    rations
    arches
    Infinitely Many Solutions
    Arch
    Lagrange multipliers
    Palais-Smale Condition
    Radially Symmetric Solutions
    Minimax
    Lagrange
    nonlinearity
    Ground State
    formulations
    ground state
    Nonlinearity
    Formulation

    Keywords

    • Deformation Theory
    • Normalized Solutions

    ASJC Scopus subject areas

    • Statistical and Nonlinear Physics
    • Mathematics(all)

    Cite this

    @article{0c4fff4ab20e44e8a7ec0d38e4d8584a,
    title = "Nonlinear Scalar Field Equations with L 2 Constraint: Mountain Pass and Symmetric Mountain Pass Approaches",
    abstract = "We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in R N {\mathbb{R}^{N}} (N ≥ 2 {N\geq 2}): { - Δ u = g (u) - μ u in R N, ≈ u ≈ L 2 (R N) 2 = m, u ∈ H 1 (R N), \displaystyle\begin{cases}-\Delta u=g(u)-\mu u\quad\text{in }\mathbb{R}^{N},{\%} \cr\lVert u\rVert-{L^{2}(\mathbb{R}^{N})}^{2}=m,\cr u\in H^{1}(\mathbb{R}^{N}){\%},\end{cases} where g (ξ) ∈ C (R, R) {g(\xi)\in C(\mathbb{R},\mathbb{R})}, m > 0 {m>0} is a given constant and μ ∈ R {\mu\in\mathbb{R}} is a Lagrange multiplier. We introduce a new approach using a Lagrange formulation of problem (∗) m {(∗)-{m}}. We develop a new deformation argument under a new version of the Palais-Smale condition. For a general class of nonlinearities related to [H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I: Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 313-345], [H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 347-375], [J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in R N {\mathbb{R}^{N}}: Mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal. 35 (2010), no. 2, 253-276], it enables us to apply minimax argument for L 2 {L^{2}} constraint problems and we show the existence of infinitely many solutions as well as mountain pass characterization of a minimizing solution of the problem inf { ∫ R N 1 2 | Δ u | 2 - G (u) d x: ≈ u ≈ L 2 (R N) 2 = m }, G (ξ) = ∫ 0 ξ g (τ) τ. \inf\Bigg{\{}\int-{\mathbb{R}^{N}}{1\over 2}|{\nabla u}|^{2}-G(u)\,dx:\lVert u{\%} \rVert-{L^{2}(\mathbb{R}^{N})}^{2}=m\Bigg{\}},\quad G(\xi)=\int-{0}^{\xi}g({\%} \tau)\,d\tau.",
    keywords = "Deformation Theory, Normalized Solutions",
    author = "Jun Hirata and Kazunaga Tanaka",
    year = "2019",
    month = "1",
    day = "1",
    doi = "10.1515/ans-2018-2039",
    language = "English",
    journal = "Advanced Nonlinear Studies",
    issn = "1536-1365",
    publisher = "Advanced Nonlinear Studies",

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    TY - JOUR

    T1 - Nonlinear Scalar Field Equations with L 2 Constraint

    T2 - Mountain Pass and Symmetric Mountain Pass Approaches

    AU - Hirata, Jun

    AU - Tanaka, Kazunaga

    PY - 2019/1/1

    Y1 - 2019/1/1

    N2 - We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in R N {\mathbb{R}^{N}} (N ≥ 2 {N\geq 2}): { - Δ u = g (u) - μ u in R N, ≈ u ≈ L 2 (R N) 2 = m, u ∈ H 1 (R N), \displaystyle\begin{cases}-\Delta u=g(u)-\mu u\quad\text{in }\mathbb{R}^{N},% \cr\lVert u\rVert-{L^{2}(\mathbb{R}^{N})}^{2}=m,\cr u\in H^{1}(\mathbb{R}^{N})%,\end{cases} where g (ξ) ∈ C (R, R) {g(\xi)\in C(\mathbb{R},\mathbb{R})}, m > 0 {m>0} is a given constant and μ ∈ R {\mu\in\mathbb{R}} is a Lagrange multiplier. We introduce a new approach using a Lagrange formulation of problem (∗) m {(∗)-{m}}. We develop a new deformation argument under a new version of the Palais-Smale condition. For a general class of nonlinearities related to [H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I: Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 313-345], [H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 347-375], [J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in R N {\mathbb{R}^{N}}: Mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal. 35 (2010), no. 2, 253-276], it enables us to apply minimax argument for L 2 {L^{2}} constraint problems and we show the existence of infinitely many solutions as well as mountain pass characterization of a minimizing solution of the problem inf { ∫ R N 1 2 | Δ u | 2 - G (u) d x: ≈ u ≈ L 2 (R N) 2 = m }, G (ξ) = ∫ 0 ξ g (τ) τ. \inf\Bigg{\{}\int-{\mathbb{R}^{N}}{1\over 2}|{\nabla u}|^{2}-G(u)\,dx:\lVert u% \rVert-{L^{2}(\mathbb{R}^{N})}^{2}=m\Bigg{\}},\quad G(\xi)=\int-{0}^{\xi}g(% \tau)\,d\tau.

    AB - We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in R N {\mathbb{R}^{N}} (N ≥ 2 {N\geq 2}): { - Δ u = g (u) - μ u in R N, ≈ u ≈ L 2 (R N) 2 = m, u ∈ H 1 (R N), \displaystyle\begin{cases}-\Delta u=g(u)-\mu u\quad\text{in }\mathbb{R}^{N},% \cr\lVert u\rVert-{L^{2}(\mathbb{R}^{N})}^{2}=m,\cr u\in H^{1}(\mathbb{R}^{N})%,\end{cases} where g (ξ) ∈ C (R, R) {g(\xi)\in C(\mathbb{R},\mathbb{R})}, m > 0 {m>0} is a given constant and μ ∈ R {\mu\in\mathbb{R}} is a Lagrange multiplier. We introduce a new approach using a Lagrange formulation of problem (∗) m {(∗)-{m}}. We develop a new deformation argument under a new version of the Palais-Smale condition. For a general class of nonlinearities related to [H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I: Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 313-345], [H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 347-375], [J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in R N {\mathbb{R}^{N}}: Mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal. 35 (2010), no. 2, 253-276], it enables us to apply minimax argument for L 2 {L^{2}} constraint problems and we show the existence of infinitely many solutions as well as mountain pass characterization of a minimizing solution of the problem inf { ∫ R N 1 2 | Δ u | 2 - G (u) d x: ≈ u ≈ L 2 (R N) 2 = m }, G (ξ) = ∫ 0 ξ g (τ) τ. \inf\Bigg{\{}\int-{\mathbb{R}^{N}}{1\over 2}|{\nabla u}|^{2}-G(u)\,dx:\lVert u% \rVert-{L^{2}(\mathbb{R}^{N})}^{2}=m\Bigg{\}},\quad G(\xi)=\int-{0}^{\xi}g(% \tau)\,d\tau.

    KW - Deformation Theory

    KW - Normalized Solutions

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