Nonlinear scalar field equations with L² constraint: Mountain pass and symmetric mountain pass approaches

We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in _R^N (N ≥ 2): (∗_)m{−∆u = g(u) − µu in _R^N , kuk_L2_(RN ₎ = m, u ∈ H¹(_R^N ), where g(ξ) ∈ C(_R, _R), m > 0 is a given constant and µ ∈ R is a Lagrange multiplier. We introduce a new approach using a Lagrange formulation of the problem (∗)_m. We develop a new deformation argument under a new version of the Palais-Smale condition. For a general class of nonlinearities related to [BL1, BL2, HIT], it enables us to apply minimax argument for L² 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{ ^Z_{RN 2}¹ |∇u|² − G(u) dx; kuk²_L2_(RN ₎ = m}, G(ξ) = ^Z₀^ξ g(τ) dτ.

35J60, 58E05