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 ${\mathbb{R}}^N$ ($N\ge 2$): $$ (*)_m \left\{ \eqalign{ -&Δu = g(u) -μu \quad \hbox{in}\ {\mathbb R}^N, \cr &\| u\|_{L^2({\mathbb R}^N)} = m, \cr &u \in H^1({\mathbb R}^N), \cr} \right. $$ where $g(\xi )\in C(\mathbb{R},\mathbb{R})$, $m>0$ is a given constant and $\mu \in \mathbb{R}$ is a Lagrange multiplier. We introduce a new approach using a Lagrange formulation of the problem $(\ast {)}_m$. 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 $L^2$ constraint problems and we show the existence of infinitely many solutions as well as mountain pass characterization of a minimizing solution of the problem: $$\inf \left\{{\int }_{{\mathbb{R}}^N}\frac{1}{2}|\nabla u{|}^2-G(u)dx;\Vert u{\Vert }_{L^2({\mathbb{R}}^N)}^2=m\right\},G(\xi )={\int }_0^{\xi }g(\tau )d\tau .$$

39 pages