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

Abstract

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} ): ${(*)_{m}}$ { - 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.