A mass supercritical problem revisited

Abstract

In any dimension $N\ge 1$ and for given mass $m>0$, we revisit the nonlinear scalar field equation with an $L^2$ constraint:

$$\begin{aligned} \left\{ \begin{aligned} -\Delta u&=f(u)-\mu u\quad \text {in}~\mathbb {R}^N,\\ \Vert u\Vert ^2_{L^2(\mathbb {R}^N)}&=m,\\ u&\in H^1(\mathbb {R}^N), \end{aligned} \right. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (P_m) \end{aligned}$$

where $\mu \in \mathbb {R}$ will arise as a Lagrange multiplier. Assuming only that the nonlinearity f is continuous and satisfies weak mass supercritical conditions, we show the existence of ground states to $(P_m)$ and reveal the basic behavior of the ground state energy $E_m$ as $m>0$ varies. In particular, to overcome the compactness issue when looking for ground states, we develop robust arguments which we believe will allow treating other $L^2$ constrained problems in general mass supercritical settings. Under the same assumptions, we also obtain infinitely many radial solutions for any $N\ge 2$ and establish the existence and multiplicity of nonradial sign-changing solutions when $N\ge 4$. Finally we propose two open problems.