Nonlinear Scalar Field Equations with L 2 Constraint: Mountain Pass and Symmetric Mountain Pass Approaches
| العنوان: | Nonlinear Scalar Field Equations with L 2 Constraint: Mountain Pass and Symmetric Mountain Pass Approaches |
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| المؤلفون: | Kazunaga Tanaka, Jun Hirata |
| المصدر: | Advanced Nonlinear Studies. 19:263-290 |
| بيانات النشر: | Walter de Gruyter GmbH, 2019. |
| سنة النشر: | 2019 |
| مصطلحات موضوعية: | 010101 applied mathematics, Combinatorics, Nonlinear system, General Mathematics, 010102 general mathematics, Lagrange formulation, Statistical and Nonlinear Physics, Nabla symbol, 0101 mathematics, Characterization (mathematics), 01 natural sciences, Scalar field, Mathematics |
| الوصف: | 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} ): ${(*)_{m}}$ { - 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 | ∇ u | 2 - G ( u ) d x : ∥ u ∥ L 2 ( ℝ N ) 2 = m } , G ( ξ ) = ∫ 0 ξ g ( τ ) 𝑑 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. |
| تدمد: | 2169-0375 1536-1365 |
| URL الوصول: | https://explore.openaire.eu/search/publication?articleId=doi_________::d5ca8abd9a9a8ae33731583e6a73e067 https://doi.org/10.1515/ans-2018-2039 |
| حقوق: | OPEN |
| رقم الأكسشن: | edsair.doi...........d5ca8abd9a9a8ae33731583e6a73e067 |
| قاعدة البيانات: | OpenAIRE |
| تدمد: | 21690375 15361365 |
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